Mark the tick against the correct answer in the following:

To Find: The value of $\operatorname{cosec}^{-1}\left(\operatorname{cosec}\left(\frac{4 \pi}{3}\right)\right)$

Now, let $x=\operatorname{cosec}^{-1}\left(\operatorname{cosec}\left(\frac{4 \pi}{3}\right)\right)$

$\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(\frac{4 \pi}{3}\right)$

Here range of principle value of cosec is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

$\Rightarrow \mathrm{x}=\frac{4 \pi}{3} \notin\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Hence for all values of $x$ in range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, the value of

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

$\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(\pi+\frac{\pi}{3}\right)\left(\because \operatorname{cosec}\left(\frac{4 \pi}{3}\right)=\operatorname{cosec}\left(\pi+\frac{\pi}{3}\right)\right)$

$\Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(-\frac{\pi}{3}\right)(\because \operatorname{cosec}(\pi+\theta)=\operatorname{cosec}(-\theta))$

$\Rightarrow x=-\frac{\pi}{3}$