91Ó°ÊÓ

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{1}{\sec (\theta)+1}+\frac{1}{\sec (\theta)-1}=2 \csc (\theta) \cot (\theta) $$

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. \(\operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right)\)

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. \(\operatorname{arccsc}\left(\csc \left(-\frac{\pi}{2}\right)\right)\)

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{1}{\csc (\theta)+1}+\frac{1}{\csc (\theta)-1}=2 \sec (\theta) \tan (\theta) $$

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. \(\operatorname{arccsc}\left(\csc \left(\frac{11 \pi}{6}\right)\right)\)

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{1}{\csc (\theta)-\cot (\theta)}-\frac{1}{\csc (\theta)+\cot (\theta)}=2 \cot (\theta) $$

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

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. \(\operatorname{arcsec}\left(\sec \left(\frac{11 \pi}{12}\right)\right)\)

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. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)

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