Q.48 The value of \sin \left(\cot ^{-1} x\right) is
A. \frac{1}{\sqrt{1+x^{2}}}
B. \sqrt{1+x^{2}}
C. \frac{1}{x \sqrt{1+x^{2}}}
D. x \sqrt{1+x^{2}}
Answer: A
Solution:
\sin \left(\cot ^{-1} x\right) \text { Let } \cot ^{-1} x=t \therefore \quad x=\cot ^{t} \therefore \quad 1+\cot ^{2} t=1+x^{2} \therefore \quad \operatorname{cosec}^{2} t=1+x^{2} \therefore \quad \operatorname{cosec} t=\sqrt{1+x^{2}} \therefore \quad \sin t=\frac{1}{\sqrt{1+x^{2}}} \therefore \quad t=\sin ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right) \therefore \quad \sin \left(\cot ^{-1} x\right)=\sin \left(\sin ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)\right) =\frac{1}{\sqrt{1+x^{2}}}