91Ó°ÊÓ

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

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

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

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

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

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) 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. $$ \frac{1-\sin (\theta)}{1+\sin (\theta)}=(\sec (\theta)-\tan (\theta))^{2} $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{11 \pi}{6}\right)\right)\)

In Exercises \(129=132\), verify the identity. You may need to consult Sections 2.2 and 6.2 for a review of the properties of absolute value and logarithms before proceeding. $$ -\ln |\csc (\theta)|=\ln |\sin (\theta)| $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)