Write the value of ^ -1 ( 3 4 ). - Vidyadip

Question

Write the value of $\cos^{-1}\Big(\tan\frac{3\pi}{4}\Big).$

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Answer

We have
$\cos^{-1}\Big(\tan\frac{3\pi}{4}\Big)=\cos^{-1}\Big\{-\tan\Big(-\pi-\frac{3\pi}{4}\Big)\Big\}$ $[\because\ \tan(\pi-\text{x}=-\tan\text{x})]$
$=\cos^{-1}\Big\{\tan\Big(-\frac{\pi}{4}\Big)\Big\}$
$=\cos^{-1}\Big\{-\tan\Big(\frac{\pi}{4}\Big)\Big\}$
$=\cos^{-1}(-1)$
$=\cos^{-1}(\cos\pi)$ $[\therefore\ \cos\pi=-1]$
$=\pi$
$\therefore\ \cos^{-1}\Big(\tan\frac{3\pi}{4}\Big)=\pi$

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