91Ó°ÊÓ

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

In Exercises \(119-130\), 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. $$ \operatorname{arccsc}\left(\csc \left(\frac{11 \pi}{6}\right)\right) $$

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

In Exercises \(119-130\), 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. $$ \operatorname{arcsec}\left(\sec \left(\frac{11 \pi}{12}\right)\right) $$

In Exercises \(119-130\), 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. $$ \operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right) $$

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. $$ -\ln |\csc (\theta)|=\ln |\sin (\theta)| $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \sin \left(\arccos \left(-\frac{1}{2}\right)\right) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \sin \left(\arccos \left(\frac{3}{5}\right)\right) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \sin (\arctan (-2)) $$

As we did in Exercise 74 in Section \(10.2\), let \(\alpha\) and \(\beta\) be the two acute angles of a right triangle. (Thus \(\alpha\) and \(\beta\) are complementary angles.) Show that \(\sec (\alpha)=\csc (\beta)\) and \(\tan (\alpha)=\cot (\beta)\). The fact that co-functions of complementary angles are equal in this case is not an accident and a more general result will be given in Section \(10.4\).