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  4. The value of cos( 2 cos-1 x + sin-1 x) at x = (1/5) is

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Q. The value of $\cos\left( 2 \cos^{-1} x + \sin^{-1} x\right) $ at $x = \frac{1}{5} $ is

Inverse Trigonometric Functions

Solution:

$\cos\left(2 \cos^{-1} x + \sin^{-1}x \right) $
$= \cos\left(\cos^{-1} x + \cos^{-1} x + \sin^{-1}x\right)$
$= \cos\left(\cos^{-1} x + \pi /2\right) =- \sin\cos^{-1} x$
$ = - \sin\sin^{-1} \sqrt{1-x^{2} } = - \sqrt{1-x^{2}} $
$= - \sqrt{1- \left(\frac{1}{5}\right)^{2}} = - \sqrt{\frac{24}{25}} = - \frac{2\sqrt{6}}{5} $