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  • Provide solution for RD Sharma math class 12 chapter 3 Inverse Trigonometric Functions exercise Very short answer question 10

Provide solution for RD Sharma math class 12 chapter 3 Inverse Trigonometric Functions exercise Very short answer question 10

Answers (1)

Answer: \frac{1}{\sqrt{x^{2}+1}}                                             

Given: \sin \left(\cot ^{-1} x\right)

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

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

Solution:

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

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

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

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

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

 

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