Question:
The value of $\cot \left(\sin ^{-1} x\right)$ is
$(a) \frac{\sqrt{1+x^{2}}}{x}$
$(\mathrm{b}) \frac{x}{\sqrt{1+x^{2}}}$
$(\mathrm{c}) \frac{1}{x}$
$(\mathrm{d}) \frac{\sqrt{1-x^{2}}}{x}$
Solution:
We know
$\sin ^{-1} x=\cot ^{-1} \frac{\sqrt{1-x^{2}}}{x}$
$\therefore \cot \left(\sin ^{-1} x\right)=\cot \left(\cot ^{-1} \frac{\sqrt{1-x^{2}}}{x}\right)$
$\Rightarrow \cot \left(\sin ^{-1} x\right)=\frac{\sqrt{1-x^{2}}}{x}$ $\left[\cot \left(\cot ^{-1} x\right)=x, \forall x \in \mathrm{R}\right]$
Thus, the value of $\cot \left(\sin ^{-1} x\right)$ is $\frac{\sqrt{1-x^{2}}}{x}$.
Hence, the correct answer is option (d).