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\))=-\(\frac\)12$

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.

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

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).