Mark the tick against the correct answer in the following: The value of $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{4 \pi}{3}\right)$ is A. $\frac{\pi}{3}$ B. $\frac{-\pi}{3}$ C. $\frac{2 \pi}{3}$ D. none of these

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Mark the tick against the correct answer in the following: The value of $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{4 \pi}{3}\right)$ is
A. $\frac{\pi}{3}$
B. $\frac{-\pi}{3}$
C. $\frac{2 \pi}{3}$
D. none of these

CBSE | Class 12 | Inverse Trigonometric Functions | Maths | RS Aggarwal

Mark the tick against the correct answer in the following: The value of $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{4 \pi}{3}\right)$ is
A. $\frac{\pi}{3}$
B. $\frac{-\pi}{3}$
C. $\frac{2 \pi}{3}$
D. none of these

Solution:

Option(B) is correct.
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 $\operatorname{cosec}$ 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
$\begin{array}{l} \Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(\pi+\frac{\pi}{3}\right)\left(\because \operatorname{cosec}\left(\frac{4 \pi}{3}\right)=\operatorname{cosec}\left(\pi+\frac{\pi}{3}\right)\right) \\ \Rightarrow \operatorname{cosec} x=\operatorname{cosec}\left(-\frac{\pi}{3}\right)(\because \operatorname{cosec}(\pi+\theta)=\operatorname{cosec}(-\theta)) \\ \Rightarrow x=-\frac{\pi}{3} \end{array}$