The value of $\cos ^{-1}\left(\cos \frac{5 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{5 \pi}{3}\right)$ is
(a) $\frac{\pi}{2}$
(b) $\frac{5 \pi}{3}$
(C) $\frac{10 \pi}{3}$
(d) 0
(d) 0
We have
$\cos ^{-1}\left(\cos \frac{5 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{5 \pi}{3}\right)=\cos ^{-1}\left\{\cos \left(2 \pi-\frac{\pi}{3}\right)\right\}+\sin ^{-1}\left\{\sin \left(2 \pi-\frac{\pi}{3}\right)\right\}$
$=\cos ^{-1}\left\{\cos \left(\frac{\pi}{3}\right)\right\}+\sin ^{-1}\left\{-\sin \left(\frac{\pi}{3}\right)\right\}$
$=\cos ^{-1}\left\{\cos \left(\frac{\pi}{3}\right)\right\}-\sin ^{-1}\left\{\sin \left(\frac{\pi}{3}\right)\right\}$
$=\frac{\pi}{3}-\frac{\pi}{3}$
$=0$
cos−1(cos5π3)+sin−1(sin5π3)=cos−1{cos(2π−π3)}+sin−1{sin(2π−π3)}=cos−1{cos(π3)}+sin−1{−sin(π3)}=cos−1{cos(π3)}−sin−1{sin(π3)}=π3−π3=0
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