Question

If \( \sin \left( \sin^{-1} \frac{1}{5} \right) + \cos^{-1} x = 1 \), find \( x \).

• A. 1
• B. 0
• C. \( \frac{4}{5} \)
• D. \( \frac{1}{5} \)

Hint:

Remember that \( \sin \left( \sin^{-1} x \right) = x \) and \( \cos^{-1} x \) represents the angle whose cosine is \( x \).

Answer 1

The Correct Option is D

We are given the equation: \[ \sin \left( \sin^{-1} \frac{1}{5} \right) + \cos^{-1} x = 1 \] Since \( \sin \left( \sin^{-1} \frac{1}{5} \right) = \frac{1}{5} \), the equation becomes: \[ \frac{1}{5} + \cos^{-1} x = 1 \] Now solve for \( \cos^{-1} x \): \[ \cos^{-1} x = 1 - \frac{1}{5} = \frac{4}{5} \] Taking the cosine of both sides: \[ x = \cos \left( \frac{4}{5} \right) \] Thus, the correct value of \( x \) is \( \frac{1}{5} \).