Evaluate the following: cosec^ -1 cosec (-9 4 ) - Vidyadip

Question

Evaluate the following:
$\text{cosec}^{-1}\Big\{\text{cosec}\Big(-\frac{9\pi}{4}\Big)\Big\}$

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Answer

$\text{cosec}^{-1}\Big\{\text{cosec}\Big(-\frac{9\pi}{4}\Big)\Big\}$
$=\text{cosec}^{-1}\Big[-\text{cosec}\Big(2\pi+\frac{\pi}{4}\Big)\Big]$
$=\text{cosec}^{-1}\Big(-\text{cosec}\frac{\pi}{4}\Big)$
$=\text{cosec}^{-1}\Big(\text{cosec}-\frac{\pi}{4}\Big)$
$=-\frac{\pi}{4}$

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