Problem
Solve: \( \sin\left(\sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x)\right) = 1 \)
Solution
Step 1: Use property of sine
\[ \sin \theta = 1 \Rightarrow \theta = \frac{\pi}{2} \]
So,
\[ \sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x) = \frac{\pi}{2} \]
Step 2: Convert using identity
\[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}x \]
Step 3: Substitute
\[ \sin^{-1}\left(\frac{1}{5}\right) + \left(\frac{\pi}{2} – \sin^{-1}(x)\right) = \frac{\pi}{2} \]
\[ \sin^{-1}\left(\frac{1}{5}\right) = \sin^{-1}(x) \]
Step 4: Final value
\[ x = \frac{1}{5} \]
Final Answer
\[ \boxed{\frac{1}{5}} \]
Explanation
Since sinθ = 1 only when θ = π/2 (within principal range), we equate angles and solve.