Question:
If \( 0 \le \cos^{-1} x \le \pi \) and \( -\frac{\pi}{2} \le \sin^{-1} x \le \frac{\pi}{2} \), then at \( x = \frac{1}{5} \) the value of \( \cos(2 \cos^{-1} x + \sin^{-1} x) \) is
If \( 0 \le \cos^{-1} x \le \pi \) and \( -\frac{\pi}{2} \le \sin^{-1} x \le \frac{\pi}{2} \), then at \( x = \frac{1}{5} \) the value of \( \cos(2 \cos^{-1} x + \sin^{-1} x) \) is
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Split $2\cos^{-1}x$ into $\cos^{-1}x + \cos^{-1}x$ to use the identity $\cos^{-1}x + \sin^{-1}x = \pi/2$.
Updated On: Apr 30, 2026
- \( -\sqrt{\frac{24}{25}} \)
- \( \sqrt{\frac{24}{25}} \)
- \( \frac{\sqrt{24}}{25} \)
- \( -\frac{\sqrt{24}}{5} \)
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The Correct Option is D
Solution and Explanation
Step 1: Simplify Argument
$\cos(\cos^{-1} x + (\cos^{-1} x + \sin^{-1} x)) = \cos(\cos^{-1} x + \pi/2)$.
Step 2: Trig Identity
$\cos(\theta + \pi/2) = -\sin \theta$.
Value $= -\sin(\cos^{-1} x)$.
Step 3: Calculate for x=1/5
If $\cos \theta = 1/5$, then $\sin \theta = \sqrt{1 - (1/5)^2} = \sqrt{24/25} = \frac{\sqrt{24}}{5}$.
Step 4: Final Value
Value $= -\frac{\sqrt{24}}{5}$.
Final Answer:(D)
$\cos(\cos^{-1} x + (\cos^{-1} x + \sin^{-1} x)) = \cos(\cos^{-1} x + \pi/2)$.
Step 2: Trig Identity
$\cos(\theta + \pi/2) = -\sin \theta$.
Value $= -\sin(\cos^{-1} x)$.
Step 3: Calculate for x=1/5
If $\cos \theta = 1/5$, then $\sin \theta = \sqrt{1 - (1/5)^2} = \sqrt{24/25} = \frac{\sqrt{24}}{5}$.
Step 4: Final Value
Value $= -\frac{\sqrt{24}}{5}$.
Final Answer:(D)
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