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Q. If $\sin ^{-1} \frac{\alpha}{17}+\cos ^{-1} \frac{4}{5}-\tan ^{-1} \frac{77}{36}=0,0<\alpha<13$, then $\sin ^{-1}(\sin \alpha)+\cos ^{-1}(\cos \alpha)$ is equal to

JEE MainJEE Main 2023Inverse Trigonometric Functions

Solution:

$ \cos ^{-1} \frac{4}{5}=\tan ^{-1} \frac{3}{4} $
$ \therefore \sin ^{-1} \frac{\alpha}{17}=\tan ^{-1} \frac{77}{36}-\tan ^{-1} \frac{3}{4}=\tan ^{-1}\left(\frac{\frac{77}{36}-\frac{3}{4}}{1+\frac{77}{36} \cdot \frac{3}{4}}\right) $
$ \sin ^{-1} \frac{\alpha}{17}=\tan ^{-1} \frac{8}{15}=\sin ^{-1} \frac{8}{17}$
$ \Rightarrow \frac{\alpha}{17}=\frac{8}{17} \Rightarrow \alpha=8$
$\therefore \sin ^{-1}(\sin 8)+\cos ^{-1}(\cos 8)$
$ =3 \pi-8+8-2 \pi $
$ =\pi$