The value of

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).

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