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

• A. \( -\sqrt{\frac{24}{25}} \)
• B. \( \sqrt{\frac{24}{25}} \)
• C. \( \frac{\sqrt{24}}{25} \)
• D. \( -\frac{\sqrt{24}}{5} \)

Hint:

Split $2\cos^{-1}x$ into $\cos^{-1}x + \cos^{-1}x$ to use the identity $\cos^{-1}x + \sin^{-1}x = \pi/2$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Use trigonometric identities to simplify the expression before substituting the value of \( x \).
Step 2: Key Formula or Approach:
Use \( \cos^{-1} x + \sin^{-1} x = \frac{\pi}{2} \).
Step 3: Detailed Explanation:
\[ \cos(2 \cos^{-1} x + \sin^{-1} x) = \cos(\cos^{-1} x + (\cos^{-1} x + \sin^{-1} x)) \] \[ = \cos(\cos^{-1} x + \frac{\pi}{2}) \] Using the identity \( \cos(A + \frac{\pi}{2}) = -\sin A \):
\[ = -\sin(\cos^{-1} x) \] We know \( \sin(\cos^{-1} x) = \sqrt{1 - x^2} \).
At \( x = \frac{1}{5} \):
\[ -\sin(\cos^{-1} \frac{1}{5}) = -\sqrt{1 - (\frac{1}{5})^2} = -\sqrt{1 - \frac{1}{25}} = -\sqrt{\frac{24}{25}} = -\frac{\sqrt{24}}{5} \] Step 4: Final Answer:
The value is \( -\frac{\sqrt{24}}{5} \).