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

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Aicharya · Accepted Answer

$\begin{aligned} & \cos \left(2 \cos ^{-1} x+\sin ^{-1} xight) \ & =\cos \left[\left(\sin ^{-1} x+\cos ^{-1} xight)+\cos ^{-1} xight] \ & =\cos \left(\frac{\pi}{2}+\cos ^{-1} xight) \ & =-\sin \left(\cos ^{-1} xight) \ & =-\sin \left(\sin ^{-1} \sqrt{\left(1-x^2ight)}ight) \ & =-\sqrt{1-x^2} \ & =-\sqrt{1-\left(\frac{1}{5}ight)^2} \ & =-\sqrt{\frac{24}{25}} \ & =-\frac{2 \sqrt{6}}{5}\end{aligned}$

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

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