Introduction to Trig Ratios and to Identity
Trig Ratios
\sin \theta = \dfrac{opp}{hyp}
\cos \theta = \dfrac{adj}{hyp}
\tan \theta = \dfrac{opp}{adj}
Reciprocal Identities
\csc \theta = \dfrac{1}{\sin \theta}
\sec \theta = \dfrac{1}{\cos \theta}
\cot \theta = \dfrac{1}{\tan \theta}
Pythagorean Identities
\sin^2 \theta + \cos^2 \theta = 1
Simplifying \cot x \sin x
ex Simplify cot \theta \times \sin \theta.
= \dfrac{\cos \theta}{\sin \theta} \times \sin \theta
= \cos \theta
Simplifying Trig Expression 1+\tan^2x
ex 1+\tan^2x
= 1+ (\dfrac{\sin \theta}{\cos \theta})^2
= 1 + \dfrac{\sin^2 \theta}{\cos^2 \theta}
= \dfrac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}
= \dfrac{1}{\cos^2 \theta}
= \sec^2 \theta
Simplifying Trig Expression \dfrac{1 + \sin \theta}{\cos^2 \theta}
ex Simplify \dfrac{1 + \sin \theta}{\cos^2 \theta}
= \dfrac{1 + \sin \theta}{1 - \sin^2 \theta}
= \dfrac{1 + \sin \theta}{(1 + \sin \theta)(1 - \sin \theta)}
= \dfrac{1}{1 - \sin \theta}
Simplifying Trig Expression
\displaystyle \begin{aligned} & \tan ^{2} \theta-\sec ^{2} \theta \=& \frac{\sin ^{2} \theta}{\cos ^{2} \theta}-\frac{1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\sin ^{2} \theta-\cos ^{2} \theta}{\cos ^{2} \theta}=-1 \end{aligned}
Introduction to Proving Identities
LS = RS
Prove that LS is the same as RS.
ex \dfrac{1}{\cos \theta} + \tan \theta = \dfrac{1 + \sin \theta}{\cos \theta}
\displaystyle \begin{aligned} L S &=\frac{1}{\cos \alpha}+\frac{\sin \alpha}{\cos \alpha} \\ &=\frac{1+\sin \alpha}{\cos \alpha}=R .5 . \end{aligned}
Proving Identity with Relative Angles
Proving Identity ex2
ex Prove 2 \cos^2 \theta + \sin^2 \theta - 1 = \cos^2 \theta
Proving Identity ex3
ex Prove \tan \theta + \dfrac{1}{\tan \theta} = \dfrac{1}{\sin \theta \cos \theta}
\displaystyle \begin{aligned} & \tan \theta+\frac{1}{\tan \theta}=\frac{1}{\sin \theta \cos \theta} \\ L S &=\frac{\sin \theta}{\cos \theta} s+\frac{\cos \theta}{\sin \theta} \frac{c}{c} \\ &=\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta} \\ &=\frac{1}{\sin \theta \cos \theta}=R S \end{aligned}
Using Pythagorean identities, state an equivalent expression.
\displaystyle
\sin^2 \theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
\tan^2 \theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
\sec^2 \theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
1 - \sin^2 \theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
1 - \csc^2 \theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
\cot^2\theta - \csc^2\theta
Using Pythagorean identities, state an equivalent expression.
\displaystyle
-\frac{1}{\csc^2\theta}
Express each expression in a simpler form.
\displaystyle
\cos \theta \sec \theta
Express each expression in a simpler form.
\displaystyle
\tan \theta \cos \theta
Express each expression in a simpler form.
\displaystyle
\tan \theta + \cot \theta
Express each expression in a simpler form.
\displaystyle
\sqrt{1 -\cos^2\theta}
Express each expression in a simpler form.
\displaystyle
\tan^2\theta -\sec^2 \theta
Express each expression in a simpler form.
\displaystyle
\frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta}
Factor to simplify each expression.
\displaystyle
\sin^4\theta +\sin^2\theta\cos^2 \theta
Factor to simplify each expression.
\displaystyle
(\sec \theta)^2(\sin \theta)^2 + (\sin \theta)^2
Factor to simplify each expression.
\displaystyle
4\cos^2\theta + 8 \cos \theta \sin \theta + 4\sin^2 \theta
Factor to simplify each expression.
\displaystyle
\sin^4\theta - \cos^4 \theta
Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.
\displaystyle
\tan \theta \cos \tan \theta =\sin \tan \theta
Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.
\displaystyle
\cot \theta \sec \theta = \csc \theta
Use the definitions of the primary trigonometric ratios in terms of x, y, and r to prove each identity.
\displaystyle
\frac{1 + \cot^2\theta}{\csc^2\theta} = 1
Prove that
\displaystyle
\cos \theta \cot \theta = \frac{1}{\sin \theta} -\sin \theta
Prove that
\displaystyle
\frac{\cos^2\theta}{1 - \sin \theta} = 1 + \sin \theta
Prove that
\displaystyle
\frac{1 + \cot \theta}{\csc \theta} = \sin \theta + \cos \theta
Prove that
\displaystyle
\frac{\tan^2\theta}{1 + \tan^2\theta} = \sin^2\theta
Prove each identity.
\displaystyle
\frac{\cos \theta \sin \theta}{\cot \theta} = 1- \cos^2\theta
Prove each identity.
\displaystyle
\frac{\sin^2\theta}{1 - \cos \theta} = 1 + \cos \theta
Prove each identity.
\displaystyle
\frac{1 + \sec \theta}{\sec \theta - 1} = \frac{1 + \cos \theta}{1 -\cos \theta}
Prove each identity.
\displaystyle
\frac{1 -\sin \theta}{1 + \sin \theta} = \frac{\csc \theta -1}{\csc \theta + 1}
Simplify
\displaystyle
\sin^2\theta + \frac{1 + \cot^2\theta}{1 + \tan^2\theta} + \cos^2\theta
.
Prove that 2\sin(-\theta) - \cot\theta \sin\theta \cos\theta = (\sin \theta -1)^2 -2
Explain why the identity
\displaystyle
\frac{\cos \theta}{1 + \sin \theta} + \frac{\cos \theta}{1 - \sin \theta} = \frac{2}{\cos \theta}
is not true for \theta = 90^o and \theta= 270^o.
Prove that
\displaystyle
\frac{1 + \sin \theta + \cos \theta}{1 -\sin \theta + \cos \theta} = \frac{1 + \sin \theta}{\cos \theta}
Prove that
\displaystyle
\frac{1}{1 + \sin \theta} + \frac{1}{1- \sin \theta} = \frac{2}{\cos^2\theta}
.
Prove that
\displaystyle
\frac{7\sin \theta + 5\cos \theta}{\cos \theta \sin \theta} = 7 \sec \theta + 5\csc \theta
.
Prove that
\displaystyle
(\csc \theta - \cot \theta)^2 = \frac{1 -\cos \theta}{1 + \cos \theta}
.
Prove that
\displaystyle
\sec^2\theta + \csc^2\theta = \frac{\csc^2\theta}{\cos^2\theta}
.
Prove that
\displaystyle
\sec\theta - \frac{\sin \theta}{\cot \theta} = \frac{1}{\sec \theta}
.
Prove that
\displaystyle
\tan^2\theta \cos^2\theta = 1 - \frac{\sin^2 \theta}{\tan^2 \theta}
.
Prove that
\displaystyle
\frac{\tan \theta - \sin \theta}{\sin^3\theta} = \frac{\sec \theta}{1 + \cos \theta}
.
Prove that
\displaystyle
\frac{1 + \sin \theta}{\tan \theta} = \cos \theta + \cot \theta
.
Prove that
\displaystyle
(\tan \theta + \cot \theta)^2 = \sec^2\theta +\frac{1}{\sin^2\theta}
.
Use a graphing calculator to graph each side to determine if the equation appears to be an identity.
\displaystyle
\frac{1}{\sin^2x} + \frac{1}{\cos^2x} = 1
Use a graphing calculator to graph each side to determine if the equation appears to be an identity.
\displaystyle
\cos x (\cos x -\sec x) = -\sin^2x
Prove that
\displaystyle
(\sin \theta + \cos \theta)^2 = \frac{2 + \sec \theta \csc \theta}{\sec \theta \csc \theta}
Prove that
\displaystyle
\frac{\cot \theta}{\csc \theta -1} + \frac{\cot \theta}{\csc \theta +1} = 2\sec \theta
Prove that
\displaystyle
\frac{\tan \theta + \cos \theta}{\sin \theta} = \frac{1}{\cos \theta} + \frac{\cos \theta}{\sin \theta}
Prove that
\displaystyle
\frac{\cos^2 \theta -\sin^2 \theta}{\cos^2 \theta + \sin \theta \cos \theta} = \frac{\cot \theta -1}{\cot \theta}
Prove that
\displaystyle
\tan \theta \sin \theta + \cos \theta - \sec \theta + 1 = \sec^2\theta \cos^2 \theta
Prove that
\displaystyle
\frac{1 -\tan^2\theta}{\tan \theta -\tan^2\theta} = 1 + \frac{1}{\tan \theta}
Prove that \tan^2\theta(1 + \cot^2\theta) = \sec^2\theta