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Question:

If $\cot \left(\cos ^{-1} \frac{3}{5}+\sin ^{-1} x\right)=0$, find the values of $x$.

Solution:

$\cot \left(\cos ^{-1} \frac{3}{5}+\sin ^{-1} x\right)=0$

$\Rightarrow \cos ^{-1} \frac{3}{5}+\sin ^{-1} x=\cot 0$

$\Rightarrow \cos ^{-1} \frac{3}{5}+\sin ^{-1} x=\frac{\pi}{2}$

$\Rightarrow \cos ^{-1} \frac{3}{5}=\frac{\pi}{2}-\sin ^{-1} x$

$\Rightarrow \cos ^{-1} \frac{3}{5}=\cos ^{-1} x \quad\left[\because \cos ^{-1} x=\frac{\pi}{2}-\sin ^{-1} x\right]$

$\Rightarrow x=\frac{3}{5}$

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