36. Q12
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Similar Question 1
<p>Use half-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>\frac{\theta}{4}</code>.</p><p><code class='latex inline'> \displaystyle \tan \frac{\theta}{4} </code></p>
Similar Question 2
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec \theta \cos \theta-\cos ^{2} \theta </code></p>
Similar Question 3
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \csc \theta, \cot \theta </code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Prove that <code class='latex inline'>2\sin(-\theta) - \cot\theta \sin\theta \cos\theta = (\sin \theta -1)^2 -2</code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \tan ^{2} \theta=(\sec \theta+1)(\sec \theta-1) </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \frac{1+\tan \theta}{\sin \theta+\cos \theta}=\sec \theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \csc \theta=1 </code></p>
<p>Prove that (sin 30<code class='latex inline'>^{\circ})^2 +</code> (cos 30<code class='latex inline'>^{\circ})^2 =</code> (sin 315<code class='latex inline'>^{\circ})^2 +</code> (cos 315<code class='latex inline'>^{\circ})^2</code>.</p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sec \theta-\tan \theta=\frac{1-\sin \theta}{\cos \theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \frac{\cos \theta \csc \theta}{\cot \theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec ^{2} \theta-1 </code></p>
<ul> <li>(a) Is this an identity? Justify your answers. </li> <li>(b) For those equations that are identities, state and restrictions on the variables.</li> </ul> <p> <code class='latex inline'> \displaystyle \frac{1 -\cos x}{\sin x} = \frac{\sin x}{1 + \cos x} </code></p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \tan \theta \cos \theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \cos \theta \sec \theta=1 </code></p>
<p>Prove that <code class='latex inline'> \displaystyle \frac{1 + \cot \theta}{\csc \theta} = \sin \theta + \cos \theta </code></p>
<p>Simplify each trigonometric expression.</p><p><code class='latex inline'>\displaystyle \tan \theta \cot \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle 1+\sec ^{2} \theta \sin ^{2} \theta=\sec ^{2} \theta </code></p>
<p>Factor to simplify each expression.</p><p><code class='latex inline'> \displaystyle \sin^4\theta +\sin^2\theta\cos^2 \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \csc ^{2} \theta-\cot ^{2} \theta=1 </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle (\cot \theta+1)^{2}=\csc ^{2} \theta+2 \cot \theta </code></p>
<p>Use double-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>4\theta</code>.</p><p><code class='latex inline'> \displaystyle \sin 4\theta </code></p>
<p>Prove that <code class='latex inline'>\tan^2\theta(1 + \cot^2\theta) = \sec^2\theta</code></p>
<p>Determine the value of each expression if <code class='latex inline'>\cos A = \dfrac{5}{14}</code> and <code class='latex inline'>\sin B = \dfrac{4}{13}</code>.</p><p><code class='latex inline'>\dfrac{\sin A \cot A}{\sec A}</code></p>
<p>Use the definitions of the primary trigonometric ratios in terms of <code class='latex inline'>x, y</code>, and <code class='latex inline'>r</code> to prove each identity.</p><p><code class='latex inline'> \displaystyle \tan \theta \cos \tan \theta =\sin \tan \theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \cot \theta=\csc \theta \cos \theta </code></p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle -\frac{1}{\csc^2\theta} </code></p>
<p>Factor to simplify each expression.</p><p><code class='latex inline'> \displaystyle 4\cos^2\theta + 8 \cos \theta \sin \theta + 4\sin^2 \theta </code></p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \sqrt{1 -\cos^2\theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \cos \theta+\sin \theta \tan \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \sec \theta \cot \theta=1 </code></p>
<p>Write an equivalent equation to the Pythagorean identity in terms of the reciprocal trigonometric ratios.</p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \sin \theta, \cos \theta </code></p>
<p>Consider an angle <code class='latex inline'>\theta</code> in the third quadrant such that <code class='latex inline'>\sec \theta = -\dfrac{a}{b}</code>. Determine expressions in terms of <code class='latex inline'>a</code> and <code class='latex inline'>b</code> for the other five trigonometric ratios for <code class='latex inline'>\theta</code>. State any restrictions on <code class='latex inline'>a</code> and <code class='latex inline'>b</code>.</p>
<p>Use half-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>\frac{\theta}{4}</code>.</p><p><code class='latex inline'> \displaystyle \tan \frac{\theta}{4} </code></p>
<p>Use a graphing calculator to graph each side to determine if the equation appears to be an identity.</p><p><code class='latex inline'>\displaystyle \frac{1}{\sin^2x} + \frac{1}{\cos^2x} = 1 </code></p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \cot \theta, \sin \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \frac{\sin \theta}{\cos \theta \tan \theta} </code></p>
<p>Simplify <code class='latex inline'>\displaystyle \tan \theta \cot \theta-\sin ^{2} \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \frac{\sec \theta}{\cot \theta+\tan \theta}=\sin \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \tan \theta(\cot \theta+\tan \theta) </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \cos ^{2} \theta=(1+\sin \theta)(1-\sin \theta) </code></p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \sec \theta, \tan \theta </code></p>
<p>Given that <code class='latex inline'>\cot \theta = \dfrac{2x}{x+1}, x \neq -1</code>, and <code class='latex inline'>\theta</code> is in the first quadrant, determine the other two reciprocal trigonometric ratios. </p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \tan \theta + \cot \theta </code></p>
<p>Use double-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>4\theta</code>.</p><p><code class='latex inline'> \displaystyle \cos 4\theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sin \theta+\cos \theta=\frac{2 \sin ^{2} \theta-1}{\sin \theta-\cos \theta} </code></p>
<p>Use half-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>\frac{\theta}{4}</code>.</p><p><code class='latex inline'> \displaystyle \sin \frac{\theta}{4} </code></p>
<p>Which expressions are equivalent?</p><p>I. <code class='latex inline'>\displaystyle (\sin \theta)(\csc \theta-\sin \theta) </code></p><p>II. <code class='latex inline'>\displaystyle \sin ^{2} \theta-1 </code></p><p>III. <code class='latex inline'>\displaystyle \cos ^{2} \theta </code></p><p>A I and II only</p><p>B II and III only</p><p>C I and III only</p><p>D I, II, and III</p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \cot \theta+\tan \theta=\frac{\sec ^{2} \theta}{\tan \theta} </code></p>
<p>Prove each identity.</p><p><code class='latex inline'> \displaystyle \frac{1 + \sec \theta}{\sec \theta - 1} = \frac{1 + \cos \theta}{1 -\cos \theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec \theta \cos \theta-\cos ^{2} \theta </code></p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \csc \theta, \cot \theta </code></p>
<p>Simplify.</p><p><code class='latex inline'> \displaystyle \sin^2\frac{\theta}{2} - \cos^2\frac{\theta}{2} </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \tan \theta=\sec \theta-\cos \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle 1-\csc ^{2} \theta </code></p>
<p>Think About a Plan Simplify the expression <code class='latex inline'>\displaystyle \frac{\tan \theta}{\sec \theta-\cos \theta} </code>.</p> <ul> <li><p>Can you write everything in terms of <code class='latex inline'>\displaystyle \sin \theta, \cos \theta </code>, or both?</p></li> <li><p>Are there any trigonometric identities that can help you simplify the expression?</p></li> </ul>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle 1 - \sin^2 \theta </code></p>
<p>Which expression is equivalent to <code class='latex inline'>\displaystyle \frac{\tan \theta}{\cos \theta - \sec \theta} </code>?</p><p>A. <code class='latex inline'>\csc \theta</code></p><p>B. <code class='latex inline'>\sec \theta</code></p><p>C. <code class='latex inline'>-\csc \theta</code></p><p>D. <code class='latex inline'>\tan^2 \theta</code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin ^{2} \theta+\cos ^{2} \theta+\tan ^{2} \theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \cos \theta \tan \theta=\sin \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \csc \theta \cos \theta \tan \theta </code></p>
<p>Prove that <code class='latex inline'> \displaystyle \frac{\tan^2\theta}{1 + \tan^2\theta} = \sin^2\theta </code></p>
<p>Determine the value of each expression if <code class='latex inline'>\cos A = \dfrac{5}{14}</code> and <code class='latex inline'>\sin B = \dfrac{4}{13}</code>.</p><p><code class='latex inline'>\dfrac{\cos A \tan A}{\csc A}</code></p>
<p>Use an angle sum identity to derive each double-angle identity.</p><p><code class='latex inline'> \displaystyle \sin 2x = 2\sin x\cos x </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \sec \theta=\tan \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \frac{\sin ^{2} \theta \csc \theta \sec \theta}{\tan \theta} </code></p>
<p>Prove each identity.</p><p><code class='latex inline'> \displaystyle \frac{\sin^2\theta}{1 - \cos \theta} = 1 + \cos \theta </code></p>
<p>Prove each identity.</p><p><code class='latex inline'> \displaystyle \frac{1 -\sin \theta}{1 + \sin \theta} = \frac{\csc \theta -1}{\csc \theta + 1} </code></p>
<img src="/qimages/27798" /><p>Given <code class='latex inline'>\displaystyle \tan \theta=\frac{3}{2} </code> and <code class='latex inline'>\displaystyle 180^{\circ} < \theta < 270^{\circ} </code>, find the exact</p><p>value of each expression.</p><p>a. <code class='latex inline'>\displaystyle \cos 2 \theta </code></p><p>b. <code class='latex inline'>\displaystyle \sin \frac{\theta}{2} </code></p><p>C. <code class='latex inline'>\displaystyle \cos \frac{\theta}{2} </code></p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle \sin^2 \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \tan \theta \csc \theta=\sec \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \csc \theta-\cos \theta \cot \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \frac{1-2 \cos ^{2} \theta}{\sin \theta \cos \theta}=\tan \theta-\cot \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin ^{2} \theta \csc \theta \sec \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \frac{\cos \theta}{1-\sin \theta}=\frac{1+\sin \theta}{\cos \theta} </code></p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle \cot^2\theta - \csc^2\theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \cot \theta(\cot \theta+\tan \theta)=\csc ^{2} \theta </code></p>
<p>Error Analysis A student simplified the expression <code class='latex inline'>\displaystyle 2-\cos ^{2} \theta </code> to <code class='latex inline'>\displaystyle 1-\sin ^{2} \theta . </code> What error did the student make? What is the correct simplified expression?</p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \csc \theta-\sin \theta=\cot \theta \cos \theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \tan \theta \cot \theta=1 </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec \theta \cos \theta \sin \theta </code></p>
<p>Determine the value of each expression if <code class='latex inline'>\cos A = \dfrac{5}{14}</code> and <code class='latex inline'>\sin B = \dfrac{4}{13}</code>.</p><p><code class='latex inline'>\dfrac{\cos B \csc B}{\tan B}</code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec ^{2} \theta \cot ^{2} \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle (\sin \theta-1)(\tan \theta+\sec \theta)=-\cos \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \frac{\csc \theta}{\sin \theta+\cos \theta \cot \theta} </code></p>
<ul> <li>(a) Is this an identity? Justify your answers. </li> <li>(b) For those equations that are identities, state and restrictions on the variables.</li> </ul> <p><code class='latex inline'> \displaystyle \frac{1 + 2\sin x \cos x}{\sin x + \cos x} = \sin x + \cos x </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \cos \theta \cot \theta=\frac{1}{\sin \theta}-\sin \theta </code></p>
<p>How can you express <code class='latex inline'>\displaystyle \csc ^{2} \theta-2 \cot ^{2} \theta </code> in terms of <code class='latex inline'>\displaystyle \sin \theta </code> and <code class='latex inline'>\displaystyle \cos \theta </code> ? <code class='latex inline'>\displaystyle \begin{array}{llll}\text { F) } \frac{1-2 \cos ^{2} \theta}{\sin ^{2} \theta} & \text { (G) } \frac{1-2 \sin ^{2} \theta}{\sin ^{2} \theta} & \text { H) } \sin ^{2} \theta-2 \cos ^{2} \theta & \text { D) } \frac{1}{\sin ^{2} \theta}-\frac{2}{\tan ^{2} \theta}\end{array} </code></p>
<p>Which expression can be used to form an identity with <code class='latex inline'>\displaystyle \frac{\tan ^{2} \theta+1}{\tan ^{2} \theta} </code> ?</p><p>A <code class='latex inline'>\displaystyle \sin ^{2} \theta </code></p><p>B <code class='latex inline'>\displaystyle \cos ^{2} \theta </code></p><p>C <code class='latex inline'>\displaystyle \tan ^{2} \theta </code></p><p><code class='latex inline'>\displaystyle \mathbf{D} \csc ^{2} \theta </code></p>
<p>Simplify.</p><p><code class='latex inline'> \displaystyle 2\cos^2\theta - \cos 2\theta </code></p>
<ul> <li>(a) Is this an identity? Justify your answers. </li> <li>(b) For those equations that are identities, state and restrictions on the variables.</li> </ul> <p><code class='latex inline'> \displaystyle \frac{\sin x}{ 1 + \cos x} = \csc x - \cot x </code></p>
<p>Vocabulary How does the identity</p><p><code class='latex inline'>\displaystyle \cos ^{2} \theta+\sin ^{2} \theta=1 </code> relate to the Pythagorean Theorem?</p>
<p><strong>a)</strong> Acute <code class='latex inline'>\triangle</code>ABC is isosceles, with <code class='latex inline'>b=c</code>. Show that <code class='latex inline'>\displaystyle{\cos A=1-\frac{a^2}{2b^2}}</code>. </p><p><strong>b)</strong> Solve <code class='latex inline'>\triangle</code>ABC, if the equal sides are 1.5 cm and the third side is 0.8 cm.</p>
<p><code class='latex inline'>\displaystyle \frac{\cot ^{2} \theta-\csc ^{2} \theta}{\tan ^{2} \theta-\sec ^{2} \theta} </code></p>
<p>Factor to simplify each expression.</p><p><code class='latex inline'> \displaystyle (\sec \theta)^2(\sin \theta)^2 + (\sin \theta)^2 </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \cos \theta=\sin \theta \cot \theta </code></p>
<p>Which expression is equivalent to <code class='latex inline'>\displaystyle 2 \cot \theta </code> ? </p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { A } \frac{1}{2 \tan \theta} & \text { (B) } \frac{2}{\cot \theta} & \text { (C) } \frac{2 \cos \theta}{\sin \theta} & \text { D) } \frac{\sin \theta}{\frac{1}{2} \cos \theta}\end{array} </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \sec \theta-\sin \theta \tan \theta=\cos \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sec \theta-\cos \theta=\tan \theta \sin \theta </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle (1-\sin \theta)(1+\sin \theta) \csc ^{2} \theta+1 </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \tan ^{2} \theta=\frac{1-\cos ^{2} \theta}{\cos ^{2} \theta} </code></p>
<ol> <li>Communication a) Select three different values for an angle <code class='latex inline'>\displaystyle x </code> between <code class='latex inline'>\displaystyle 0^{\circ} </code> and <code class='latex inline'>\displaystyle 90^{\circ} . </code> For each value of <code class='latex inline'>\displaystyle x </code>, evaluate the expression <code class='latex inline'>\displaystyle (\sin x)^{2}+(\cos x)^{2} </code></li> </ol> <p>b) Use the results from part a) to write a conjecture.</p><p>c) Verify your conjecture from part b) by simplifying</p><p>$\displaystyle </p><p>\left(\frac{\text { opposite }}{\text { hypotenuse }}\right)^{2}+\left(\frac{\text { adjacent }}{\text { hypotenuse }}\right)^{2}</p><p>$</p>
<ul> <li>(a) Is this an identity? Justify your answers. </li> <li>(b) For those equations that are identities, state and restrictions on the variables.</li> </ul> <p><code class='latex inline'> \displaystyle \dfrac{\sin x\tan x}{\sin x + \tan x} = \sin x \tan x </code></p>
<p>Use the definitions of the primary trigonometric ratios in terms of <code class='latex inline'>x, y</code>, and <code class='latex inline'>r</code> to prove each identity.</p><p><code class='latex inline'> \displaystyle \cot \theta \sec \theta = \csc \theta </code></p>
<p>Verify that each equation is an identity. <code class='latex inline'>\displaystyle \cos \theta \cos (-\theta)-\sin \theta \sin (-\theta)=1 </code></p>
<p>Use expressions for the primary trigonometric ratios in terms of <code class='latex inline'>x, y</code>, and <code class='latex inline'>r</code> to show that <code class='latex inline'>\sin^2 \theta + \cos^2 \theta = 1,</code> regardless of the value of <code class='latex inline'>\theta</code>. This equation is known as the Pythagorean identity. Why is this name appropriate?</p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle 1 - \csc^2 \theta </code></p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \cos \theta \sec \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \cos \theta(\tan \theta+\cot \theta)=1 </code></p>
<p>Which equation is NOT an identity?</p><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { F) } \cos ^{2} \theta=1-\sin ^{2} \theta & \text { H) } \sin ^{2} \theta=\cos ^{2} \theta-1 \ \text { (G) } \cot ^{2} \theta=\csc ^{2} \theta-1 & \text { D) } \tan ^{2} \theta=\sec ^{2} \theta-1\end{array} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sec ^{2} \theta-\tan ^{2} \theta </code></p>
<p>Prove that <code class='latex inline'> \displaystyle \cos \theta \cot \theta = \frac{1}{\sin \theta} -\sin \theta </code></p>
<p>Show that for any <code class='latex inline'>\angle A</code>, <code class='latex inline'>(\sin A)^2+(\cos A)^2=1</code></p>
<p>Use the definitions of the primary trigonometric ratios in terms of <code class='latex inline'>x, y</code>, and <code class='latex inline'>r</code> to prove each identity.</p><p><code class='latex inline'> \displaystyle \frac{1 + \cot^2\theta}{\csc^2\theta} = 1 </code></p>
<p>Determine the value of each expression if <code class='latex inline'>\cos A = \dfrac{5}{14}</code> and <code class='latex inline'>\sin B = \dfrac{4}{13}</code>.</p><p><code class='latex inline'>\dfrac{\sin B \sec B}{\cot B}</code></p>
<p>Prove each identity.</p><p><code class='latex inline'> \displaystyle \frac{\cos \theta \sin \theta}{\cot \theta} = 1- \cos^2\theta </code></p>
<p>Use the Tangent Half-Angle Identity and a Pythagorean identity to prove each identity.</p><p><code class='latex inline'> \displaystyle \tan \frac{A}{2} = \frac{1 -\cos A}{\sin A} </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sin \theta=\frac{\sec \theta}{\tan \theta+\cot \theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin \theta\left(1+\cot ^{2} \theta\right) </code></p>
<p>Explain why the identity</p><p><code class='latex inline'> \displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} = \frac{2}{\cos \theta} </code> is not true for <code class='latex inline'>\theta = 90^o</code> and <code class='latex inline'>\theta= 270^o</code>.</p>
<p>Express <code class='latex inline'>\displaystyle \cos \theta \csc \theta \cot \theta </code> in terms of <code class='latex inline'>\displaystyle \sin \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \sin ^{2} \theta \tan ^{2} \theta=\tan ^{2} \theta-\sin ^{2} \theta </code></p>
<p>Prove that <code class='latex inline'> \displaystyle \frac{\cos^2\theta}{1 - \sin \theta} = 1 + \sin \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin \theta \sec \theta \cot \theta </code></p>
<p>Prove each identity by writing all trigonometric ratios in terms of <code class='latex inline'>x</code>, <code class='latex inline'>y</code>, and <code class='latex inline'>r</code>. State the restrictions on <code class='latex inline'>\theta</code>.</p><p><code class='latex inline'> \displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta} </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin \theta \csc \theta </code></p>
<p>Show that <code class='latex inline'>\sec x + 1(\sec x -1) = \tan^2x</code> is an identity.</p>
<p>Factor to simplify each expression.</p><p><code class='latex inline'> \displaystyle \sin^4\theta - \cos^4 \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \csc ^{2} \theta\left(1-\cos ^{2} \theta\right) </code></p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \cot \theta, \csc \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \tan \theta=\frac{\sec \theta}{\csc \theta} </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \tan ^{2} \theta \csc ^{2} \theta=1+\tan ^{2} \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \frac{1-\cos \theta}{1+\cos \theta}=(\csc \theta-\cot \theta)^{2} </code></p>
<p>Use double-angle identities to write each expression, using trigonometric functions of <code class='latex inline'>\theta</code> instead of <code class='latex inline'>4\theta</code>.</p><p><code class='latex inline'> \displaystyle \tan 4\theta </code></p>
<p>Simplify.</p><p><code class='latex inline'> \displaystyle \frac{\cos 2\theta}{\sin \theta + \cos \theta} </code></p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle \tan^2 \theta </code></p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \tan^2\theta -\sec^2 \theta </code></p>
<p>Prove that</p><p><code class='latex inline'>\displaystyle \frac{1 -\tan^2\theta}{\tan \theta -\tan^2\theta} = 1 + \frac{1}{\tan \theta} </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle \sec \theta \csc \theta=\tan \theta+\cot \theta </code></p>
<p>Verify the identity.</p><p><code class='latex inline'>\displaystyle \frac{1-\sin \theta}{\cos \theta}=\frac{\cos \theta}{1+\sin \theta} </code></p>
<p>Verify that each equation is an identity. <code class='latex inline'>\displaystyle \cos ^{2} \theta+\tan ^{2} \theta \cos ^{2} \theta=1 </code></p>
<p>Prove each identity by writing all trigonometric ratios in terms of <code class='latex inline'>x</code>, <code class='latex inline'>y</code>, and <code class='latex inline'>r</code>. State the restrictions on <code class='latex inline'>\theta</code>.</p><p><code class='latex inline'> \displaystyle \tan\theta \cos\theta = \sin\theta </code></p>
<p>Verify each identity. Give the domain of validity for each identity.</p><p><code class='latex inline'>\displaystyle \sin \theta \cot \theta=\cos \theta </code></p>
<p>Using Pythagorean identities, state an equivalent expression.</p><p><code class='latex inline'> \displaystyle \sec^2 \theta </code></p>
<p>Simplify.</p><p><code class='latex inline'>\displaystyle 1-\cos ^{2} \theta </code></p>
<p>Verify that each equation is an identity.</p><p><code class='latex inline'>\displaystyle (\sin \theta+\cos \theta)^{2}=\frac{2+\sec \theta \csc \theta}{\sec \theta \csc \theta} </code></p>
<p>Express the first trigonometric function in terms of the second</p><p><code class='latex inline'>\displaystyle \tan \theta, \cos \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \sin \theta \cot \theta </code></p>
<p>Simplify the trigonometric expression.</p><p><code class='latex inline'>\displaystyle \cos \theta \tan \theta </code></p>
<p>Express <code class='latex inline'>\displaystyle \frac{\cos \theta}{\sec \theta+\tan \theta} </code> in terms of <code class='latex inline'>\displaystyle \sin \theta </code></p>
<p>Use a graphing calculator to graph each side to determine if the equation appears to be an identity.</p><p><code class='latex inline'>\displaystyle \cos x (\cos x -\sec x) = -\sin^2x </code></p>
<p>Express each expression in a simpler form.</p><p><code class='latex inline'> \displaystyle \frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} </code></p>
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