91Ó°ÊÓ

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \tan (\theta) \cot (\theta)=1 $$

In Exercises \(81-86\), solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-\pi \leq x \leq \pi\). $$ \cos (x) \geq \sin (x) $$

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

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \csc (\theta) \cos (\theta)=\cot (\theta) $$

In Exercises \(87-92,\) solve the inequality. Express the exact answer in interval notation, restricting your attention to \(-2 \pi \leq x \leq 2 \pi\). $$ \csc (x)>1 $$

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{\sin (\theta)}{\cos ^{2}(\theta)}=\sec (\theta) \tan (\theta) $$

Find the exact value or state that it is undefined. $$ \arcsin \left(\sin \left(\frac{\pi}{6}\right)\right) $$

In Exercises \(82-128\), 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) \(\cos (2 \theta)=\frac{1-x^{2}}{x^{2}+1}\)

Find the exact value or state that it is undefined. $$ \arcsin \left(\sin \left(-\frac{\pi}{3}\right)\right) $$