The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$, where $0 \leq \cos ^{-1} x \leq \pi$ and $-\frac{\pi}{2} \leq \sin ^{-1} x \leq \frac{\pi}{2}$, is

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Aicharya · Accepted Answer

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

The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$, where $0 \leq \cos ^{-1} x…

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