Question
For the principal values, evaluate the following:
$\sec^{-1}\big(\sqrt2\big)+2\text{cosec}^{-1}\big(-\sqrt2\big)$
$\sec^{-1}\big(\sqrt2\big)+2\text{cosec}^{-1}\big(-\sqrt2\big)$
✓
Answer
$\sec^{-1}\big(\sqrt2\big)+2\text{cosec}^{-1}\big(-\sqrt2\big)$
$\sec^{-1}\Big(\sec\frac{\pi}{4}\Big)+2\text{cosec}^{-1}\Big[\text{cosec}\Big(-\frac{\pi}{4}\Big)\Big]$
$=\frac{\pi}{4}-2\times\frac{\pi}{4}$
$=\frac{\pi}{4}-\frac{\pi}{2}$
$=-\frac{\pi}{4}$
$\sec^{-1}\Big(\sec\frac{\pi}{4}\Big)+2\text{cosec}^{-1}\Big[\text{cosec}\Big(-\frac{\pi}{4}\Big)\Big]$
$=\frac{\pi}{4}-2\times\frac{\pi}{4}$
$=\frac{\pi}{4}-\frac{\pi}{2}$
$=-\frac{\pi}{4}$
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