#### Answer

B

#### Work Step by Step

A. From the definition of arctan,
x is the number from $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ for which $\tan x=-1$
$-\displaystyle \frac{\pi}{2}\not\in (-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}) $ so
$-\displaystyle \frac{\pi}{2}$ is NOT the solution .
B. From the definition of arcsin,
x is the number from $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ for which $\sin x=-1.$
$-\displaystyle \frac{\pi}{2}\in [-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ and $\sin x=-1.$ so
$-\displaystyle \frac{\pi}{2}$ is the solution.
C. From the definition of arccos,
x is the number from $[0, \pi]$ for which $\cos x=-1.$
$-\displaystyle \frac{\pi}{2}\in [0, \pi] $and $\displaystyle \cos(-\frac{\pi}{2})=0.$ so
$-\displaystyle \frac{\pi}{2}$ is NOT the solution.