If \cos^{-1}x=\sin^{-1}x, then which one of the following is correct?
- A. x=1
- B. x=\frac{1}{2}
- C. x=\frac{1}{\sqrt{2}} ✓
- D. x=\frac{1}{\sqrt{3}}
Correct Answer: C. x=\frac{1}{\sqrt{2}}
Explanation
We know the identity \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}. Given that \cos^{-1}x = \sin^{-1}x, we substitute to get 2\sin^{-1}x = \frac{\pi}{2} \Rightarrow \sin^{-1}x = \frac{\pi}{4}. Therefore, x = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}.