The value of \(\cos(2\cos^{-1}x + \sin^{-1}x)\) at \(x =\displaystyle\frac{1}{5}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-\sqrt{\displaystyle\frac{24}{25}}\)
Let \(\theta = \cos^{-1}x\), so \(\theta\) lies in \([0, \pi]\) and \(\sin\theta \ge 0\). By the identity \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\):
\(\displaystyle 2\cos^{-1}x + \sin^{-1}x = \cos^{-1}x + \frac{\pi}{2} = \theta + \frac{\pi}{2}\)
\(\displaystyle \cos\left(\theta + \frac{\pi}{2}\right) = -\sin\theta\)
\(\displaystyle = -\sqrt{1 - x^2}\)
At \(x = \frac{1}{5}\):
\(\displaystyle -\sqrt{1 - \frac{1}{25}} = -\sqrt{\frac{24}{25}}\)
Note: This topic was not in the CUET 2026 syllabus.