Inverse Circular Functions 2 Question 13
13. Prove that $\cos \tan ^{-1}\left[\sin \left(\cot ^{-1} x\right)\right]=\sqrt{\frac{x^{2}+1}{x^{2}+2}}$.
(2002, 5M)
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Answer:
Correct Answer: 13. 3 14. $\frac{-2 \sqrt{6}}{5}$
Solution:
- LHS $=\cos \tan ^{-1}\left[\sin \left(\cot ^{-1} x\right)\right]$
$$ \begin{aligned} & =\cos \tan ^{-1} \sin \sin ^{-1} \frac{1}{\sqrt{1+x^{2}}} \\ & =\cos \tan ^{-1} \frac{1}{\sqrt{1+x^{2}}}=\sqrt{\frac{x^{2}+1}{x^{2}+2}}=\text { RHS } \end{aligned} $$
