Write the value

We have

$\cos ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\cos ^{-1}\left\{-\tan \left(\pi-\frac{3 \pi}{4}\right)\right\} \quad[\because \tan (\pi-x)=-\tan x]$

$=\cos ^{-1}\left\{\tan \left(-\frac{\pi}{4}\right)\right\}$

$=\cos ^{-1}\left\{-\tan \left(\frac{\pi}{4}\right)\right\}$

$=\cos ^{-1}(-1)$

$=\cos ^{-1}(\cos \pi) \quad[\because \cos \pi=-1]$

$=\pi$

$\therefore \cos ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\pi$