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# Find the principal values of the following: $cosec^{-1} (-\sqrt 2)$

$\begin{array}{1 1} \large {\frac{-\pi}{4}} \\ \large {\frac{-\pi}{2}} \\ \large {\frac{-\pi}{8}} \\ \large {\frac{-\pi}{6}} \end{array}$

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Ans : $$cosec^{-1}cosec \bigg( \large\frac{-\pi}{4} \bigg) = \large\frac{-\pi}{4}$$

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• The range of the principal value of $\; cosec^{-1}x$ is $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Let $cosec^{-1}-\sqrt2 = x \Rightarrow cosec (x) =-\sqrt 2$
The range of the principal value of $\; cosec^{-1}x$ is $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
$$\therefore$$ $cosec (x) = -\sqrt 2 = cosec \large\frac{-\pi}{4}$
$\Rightarrow x=\large\frac{-\pi}{4}$, where $x \;\epsilon\;$ $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ] -${$0$}
Hence the principal value of $\; cosec^{-1}(-\sqrt 2)$ is $\large\frac{-\pi}{4}$