Problem
If \( \cos^{-1}x + \cos^{-1}y = \frac{\pi}{4} \), find \( \sin^{-1}x + \sin^{-1}y \).
Solution
Step 1: Use identity
\[ \sin^{-1}t + \cos^{-1}t = \frac{\pi}{2} \]
So,
\[ \sin^{-1}x = \frac{\pi}{2} – \cos^{-1}x \]
\[ \sin^{-1}y = \frac{\pi}{2} – \cos^{-1}y \]
Step 2: Add both
\[ \sin^{-1}x + \sin^{-1}y = \left(\frac{\pi}{2} – \cos^{-1}x\right) + \left(\frac{\pi}{2} – \cos^{-1}y\right) \]
\[ = \pi – \left(\cos^{-1}x + \cos^{-1}y\right) \]
Step 3: Substitute given value
\[ = \pi – \frac{\pi}{4} = \frac{3\pi}{4} \]
Final Answer
\[ \boxed{\frac{3\pi}{4}} \]
Explanation
Using sin⁻¹x + cos⁻¹x = π/2, we convert the expression and substitute the given value.