91Ó°ÊÓ

Verify the identity. Assume that all quantities are defined. $$ \cos (\theta)(\tan (\theta)+\cot (\theta))=\csc (\theta) $$

Verify the identity. Assume that all quantities are defined. $$ \sin (\theta)(\tan (\theta)+\cot (\theta))=\sec (\theta) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{\pi}{6}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1}{1-\cos (\theta)}+\frac{1}{1+\cos (\theta)}=2 \csc ^{2}(\theta) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{5 \pi}{4}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1}{\sec (\theta)+1}+\frac{1}{\sec (\theta)-1}=2 \csc (\theta) \cot (\theta) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(-\frac{\pi}{2}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1}{\csc (\theta)+1}+\frac{1}{\csc (\theta)-1}=2 \sec (\theta) \tan (\theta) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1}{\csc (\theta)-\cot (\theta)}-\frac{1}{\csc (\theta)+\cot (\theta)}=2 \cot (\theta) $$