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  4. sin[cot-1 cos(tan-1 :x) ] =

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Q. $sin\left[cot^{-1}\left\{cos\left(tan^{-1}\:x\right)\right\}\right] = $

Inverse Trigonometric Functions

Solution:

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