Evaluate each of the following:

Solution:

We know that

$\operatorname{cosec}^{-1}(\operatorname{cosec} \theta)=\theta, \quad[-\pi / 2,0) \cup(0, \pi / 2]$

(i) $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{\pi}{4}\right)=\frac{\pi}{4}$

(ii)

$\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{3 \pi}{4}\right)=\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(\pi-\frac{\pi}{4}\right)\right]$

$=\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{\pi}{4}\right)$

$=\frac{\pi}{4}$

(iii)

$\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{6 \pi}{5}\right)=\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(\pi+\frac{\pi}{5}\right)\right]$

$=\operatorname{cosec}^{-1}\left(\operatorname{cosec}-\frac{\pi}{5}\right)$

$=-\frac{\pi}{5}$

(iv)

$\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{11 \pi}{6}\right)=\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(2 \pi-\frac{\pi}{6}\right)\right]$

$=\operatorname{cosec}^{-1}\left(\operatorname{cosec}-\frac{\pi}{6}\right)$

$=-\frac{\pi}{6}$

(v)

$\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{13 \pi}{6}\right)=\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(2 \pi+\frac{\pi}{6}\right)\right]$

$=\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{\pi}{6}\right)$

$=\frac{\pi}{6}$

(vi)

$\operatorname{cosec}^{-1}\left\{\operatorname{cosec}\left(-\frac{9 \pi}{4}\right)\right\}=\operatorname{cosec}^{-1}\left[-\operatorname{cosec}\left(2 \pi+\frac{\pi}{4}\right)\right]$

$=\operatorname{cosec}^{-1}\left(-\operatorname{cosec} \frac{\pi}{4}\right)$

$=\operatorname{cosec}^{-1}\left(\operatorname{cosec}-\frac{\pi}{4}\right)$

$=-\frac{\pi}{4}$