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Question:

If $3 \sin ^{-1} x=\pi-\cos ^{-1} x$, then $x=$  __________________.

Solution:

$3 \sin ^{-1} x=\pi-\cos ^{-1} x$

$\Rightarrow 3\left(\frac{\pi}{2}-\cos ^{-1} x\right)=\pi-\cos ^{-1} x$                   $\left(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right)$

$\Rightarrow \frac{3 \pi}{2}-3 \cos ^{-1} x=\pi-\cos ^{-1} x$

$\Rightarrow 2 \cos ^{-1} x=\frac{3 \pi}{2}-\pi=\frac{\pi}{2}$

$\Rightarrow \cos ^{-1} x=\frac{\pi}{4}$

$\Rightarrow x=\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}$

If $3 \sin ^{-1} x=\pi-\cos ^{-1} x$, then $x=\frac{1}{\sqrt{2}}$

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