0. Functions

Simplify the expression.

$\tan^2\theta-\sec^2\theta+1$

Simplify the expression.

$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

Simplify the expression.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)\csc^2\left(\theta\right)\cos^2\left(-\theta\right)$

Identify the most helpful first step in verifying the identity.

$\left(\frac{\tan^2\theta}{\sin^2\theta}-1\right)=\sec^2\theta\sin^2\left(-\theta\right)$

Identify the most helpful first step in verifying the identity.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Find all solutions to the equation.

$\left(\cos\theta+\sin\theta\right)\left(\cos\theta-\sin\theta\right)=-\frac12$

Find all solutions to the equation where 0 ≤ $\theta$ ≤ $2\pi$.

$\sin\theta\cos\left(2\theta\right)-\sin\left(2\theta\right)\cos\theta=\frac{\sqrt2}{2}$

Prove the following identities.

$\sec\theta=\frac{1}{\cos\theta}$

Prove the following identities.

$\tan\theta=\frac{\sin\theta}{\cos\theta}$

Prove the following identities.

$\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$

Prove the following identities.

$\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$

What are the three Pythagorean identities for the trigonometric functions?

For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.

cos 2θ + cos θ = 0

Derive a formula for tan (A − B).

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).