91Ó°ÊÓ

Verify the identity. Assume that all quantities are defined. $$ \frac{\cos (\theta)}{\sin ^{2}(\theta)}=\csc (\theta) \cot (\theta) $$

Suppose \(\theta\) is a Quadrant I angle with \(\tan (\theta)=x\). Verify the following formulas (a) \(\cos (\theta)=\frac{1}{\sqrt{x^{2}+1}}\) (b) \(\sin (\theta)=\frac{x}{\sqrt{x^{2}+1}}\) (c) \(\sin (2 \theta)=\frac{2 x}{x^{2}+1}\) (d) \(\cos (2 \theta)=\frac{1-x^{2}}{x^{2}+1}\)

Solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-2 \pi \leq x \leq 2 \pi\). $$ \cos (x) \leq \frac{5}{3} $$

In Exercises \(87-106\), find the exact value or state that it is undefined. $$ \arcsin \left(\sin \left(-\frac{\pi}{3}\right)\right) $$

In Exercises \(87-106\), find the exact value or state that it is undefined. $$ \arcsin \left(\sin \left(\frac{3 \pi}{4}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1+\sin (\theta)}{\cos (\theta)}=\sec (\theta)+\tan (\theta) $$

Solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-2 \pi \leq x \leq 2 \pi\). $$ \cot (x) \geq 5 $$

If \(\sin (\theta)=\frac{x}{2}\) for \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\), find an expression for \(\cos (2 \theta)\) in terms of \(x\).

Verify the identity. Assume that all quantities are defined. $$ \frac{1-\cos (\theta)}{\sin (\theta)}=\csc (\theta)-\cot (\theta) $$

Solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-2 \pi \leq x \leq 2 \pi\). $$ \tan ^{2}(x) \geq 1 $$