Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.

\displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta}

Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.

\displaystyle \tan\theta \cos\theta = \sin\theta

Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.

\displaystyle \text{csc }\theta = \frac{1}{\text{sin }\theta}

Prove each identity by writing all trigonometric ratios in terms of x, y, and r. State the restrictions on \theta.

\displaystyle \text{cos }\theta \text{ sec }\theta = 1

Simplify each expression.

\displaystyle (1 - \text{sin }\alpha)(1 + \text{sin }\alpha)

Simplify each expression.

\displaystyle \frac{\text{tan }\alpha}{\text{sin }\alpha}

Simplify each expression.

\displaystyle \text{cos}^{2}\alpha + \text{sin}^{2}\alpha

Simplify each expression.

\displaystyle \text{cot }\alpha \text{ sin }\alpha

Factor each expression.

\displaystyle 1 - \text{cos}^{2}\theta

Factor each expression.

\displaystyle \text{sin}^{2}\theta - \text{cos}^{2}\theta

Factor each expression.

\displaystyle \text{sin}^{2}\theta - 2\text{sin }\theta + 1

Factor each expression.

\displaystyle \text{cos }\theta - cos^{2}\theta

Prove that \frac{\cos^{2}\phi}{1 - \sin\phi} = 1 + \sin\phi, where \sin\phi \neq 1, by expressing \cos^{2}\phi in terms of \sin \phi.

Prove the identity. State any restrictions on the variables.

\displaystyle \frac{\sin x}{\tan x} = \cos x

Prove the identity. State any restrictions on the variables.

\displaystyle \frac{\tan\theta}{\cos\theta} = \frac{\sin\theta}{1 - \sin^{2}\theta}

Prove the identity. State any restrictions on the variables.

\displaystyle \frac{1}{\cos\alpha} + \tan\alpha = \frac{1 + \sin\alpha}{\cos\alpha}

Prove the identity. State any restrictions on the variables.

\displaystyle 1 - \cos^{2}\theta = \sin\theta\cos\theta\tan\theta

Mark claimed that \displaystyle \frac{1}{\cot \theta} = \tan \theta is an identity. Nancy let \theta = 30^o and found that both sides of the equation worked out to 1 . She said that this proves that the equation is an identity. Is Nancy's reasoning correct? Explain.

Simplify each trigonometric expression.

\displaystyle \sin \theta \cot \theta - \sin \theta \cos \theta

Simplify each trigonometric expression.

\displaystyle \cos\theta(1 + \sec\theta)(\cos\theta - 1)

Simplify each trigonometric expression.

\displaystyle (\sin x +\cos x)(\sin x -\cos x) + 2\cos^2x

Simplify each trigonometric expression.

\displaystyle \frac{\csc^{2}\theta - 3\csc\theta + 2}{\csc^{2}\theta - 1}

Prove each identity. State any restrictions on the variables.

\dfrac{\sin^{2}\phi}{1 - \cos\phi} = 1 + \cos\phi

Prove each identity. State any restrictions on the variables.

\dfrac{\tan^{2}\alpha}{1 + \tan^{2}\alpha} = \sin^{2}\alpha

Prove each identity. State any restrictions on the variables.

\cos^{2}x = (1 - \sin x)(1 + \sin x)

Prove each identity. State any restrictions on the variables.

\sin^{2}\theta + 2\cos^{2}\theta - 1 = \cos^{2}\theta

Prove each identity. State any restrictions on the variables.

\sin^{4}\alpha - cos^{4}\alpha = \sin^{2}\alpha - \cos^{2}\alpha

Prove each identity. State any restrictions on the variables.

\tan\theta + \dfrac{1}{\tan\theta} = \dfrac{1}{\sin\theta\cos\theta}

Is \csc^{2}\theta + \sec^{2}\theta = 1 an identity? Prove that it is true or demonstrate why it is false.

Prove that \sin^{2}x(1 + \dfrac{1}{\tan^{2}x}) = 1, where \sin x \neq 0.

Prove the identity. State any restrictions on the variables.

\displaystyle \frac{\sin^2x + 2\cos x - 1}{\sin^2 x +3\cos x -3} = \frac{\cos^2x + \cos x}{-\sin^2 x}

Prove the identity. State any restrictions on the variables.

\displaystyle \sin^2x -\cos^2x -\tan^2x = \frac{2\sin^2x -2\sin^4 x -1}{1- \sin^2 x}

Show how you can create several new identities from the identity \sin^2x + \cos^2 x=1 by adding, subtracting, multiplying, or dividing both sides of the equation by the same expression.

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

(1 - \cos^{2}x)(1 - \tan^{2}x) = \frac{\sin^{2}x - 2\sin^{4}x} {1 - \sin^{2}x}

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

1 - 2\cos^{2}\phi = \sin^{4}\phi - \cos^{4}\phi

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

\displaystyle \dfrac{\sin x\tan x}{\sin x + \tan x} = \sin x \tan x

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

\displaystyle \frac{1 + 2\sin x \cos x}{\sin x + \cos x} = \sin x + \cos x

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

\displaystyle \frac{1 -\cos x}{\sin x} = \frac{\sin x}{1 + \cos x}

• (a) Is this an identity? Justify your answers.
• (b) For those equations that are identities, state and restrictions on the variables.

\displaystyle \frac{\sin x}{ 1 + \cos x} = \csc x - \cot x

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}