Question

Let $f(\alpha)=\sin ^{-1}(\sin \alpha)+\cos ^{-1}(\cos \alpha)$ and $g(\beta)=\sin ^{-1}(\sin \beta)+\tan ^{-1}(\tan \beta)$, then find the value of $f(100)+g(8)$.

Answer 1

$f (100)=\sin ^{-1} \sin 100+\cos ^{-1} \cos 100$
$\sin ^{-1} \sin (100-32 \pi)+\cos ^{-1} \cos (32 \pi-100)=100-32 \pi+32 \pi-100=0 $
$g (8)=\sin ^{-1} \sin 8+\tan ^{-1} \tan 8=\sin ^{-1} \sin (3 \pi-8)+\tan ^{-1} \tan (8-3 \pi)=0$
$\therefore f (100)+ g (8)=0 $