Write the value

Question:

Write the value of $\sin \left(\cot ^{-1} x\right)$.

 

Solution:

We know

$\cot ^{-1} x=\tan ^{-1} \frac{1}{x}$

Now, we have

$\sin \left(\cot ^{-1} x\right)=\sin \left(\tan ^{-1} \frac{1}{x}\right)$

$=\sin \left[\sin ^{-1}\left(\frac{\frac{1}{x}}{\sqrt{1+\frac{1}{x^{2}}}}\right)\right] \quad\left[\because \tan ^{-1} x=\sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right)\right]$

$=\sin \left[\sin ^{-1}\left(\frac{\frac{1}{x}}{\frac{\sqrt{x^{2}+1}}{x}}\right)\right]$

$=\sin \left(\sin ^{-1} \frac{1}{\sqrt{x^{2}+1}}\right)$

$=\frac{1}{\sqrt{x^{2}+1}} \quad\left[\because \sin \left(\sin ^{-1} x=x\right)\right]$

Hence, $\sin \left(\cot ^{-1} x\right)=\frac{1}{\sqrt{x^{2}-1}}$

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