Question:
If $4 \sin ^{-1} x+\cos ^{-1} x=\pi$, then what is the value of $x$ ?
Solution:
We know that $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$
$\therefore 4 \sin ^{-1} x+\cos ^{-1} x=\pi$
$\Rightarrow 4 \sin ^{-1} x+\frac{\pi}{2}-\sin ^{-1} x=\pi \quad\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$
$\Rightarrow 3 \sin ^{-1} x=\frac{\pi}{2}$
$\Rightarrow \sin ^{-1} x=\frac{\pi}{6}$
$\Rightarrow x=\sin \frac{\pi}{6}$
$\Rightarrow x=\frac{1}{2}$
$\therefore x=\frac{1}{2}$
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