Q79 / 125 · BITSAT 2021 · Mathematics
The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ is
- Correct answerA$-\frac{2 \sqrt{6}}{5}$
- B$-2 \sqrt{6}$
- C$-\frac{\sqrt{6}}{5}$
- DNone of these
Solution
$\cos \left[2 \cos ^{-1} x+\sin ^{-1} x\right]$
$=\cos \left[\cos ^{-1} x+\cos ^{-1} x+\sin ^{-1} x\right]$
$=\cos \left[\cos ^{-1} x+\frac{\pi}{2}\right]=-\sin \left[\cos ^{-1} x\right]$
$=-\sin \left[\sin ^{-1} \sqrt{1-x^{2}}\right]=-\sqrt{1-x^{2}}$
$=-\sqrt{1-\left(\frac{1}{5}\right)^{2}}=-\sqrt{\frac{24}{25}}=-\frac{2 \sqrt{6}}{5}$
Hence, the correct answer is option: 'A'.