Problem
Find x if \( \cot\left(\cos^{-1}\left(\frac{3}{5}\right) + \sin^{-1}(x)\right) = 0 \)
Solution
Step 1: Use property of cot
\[ \cot \theta = 0 \Rightarrow \theta = \frac{\pi}{2} \]
So,
\[ \cos^{-1}\left(\frac{3}{5}\right) + \sin^{-1}(x) = \frac{\pi}{2} \]
Step 2: Convert using identity
\[ \cos^{-1}t = \frac{\pi}{2} – \sin^{-1}t \]
\[ \cos^{-1}\left(\frac{3}{5}\right) = \frac{\pi}{2} – \sin^{-1}\left(\frac{3}{5}\right) \]
Step 3: Substitute
\[ \left(\frac{\pi}{2} – \sin^{-1}\left(\frac{3}{5}\right)\right) + \sin^{-1}(x) = \frac{\pi}{2} \]
\[ \sin^{-1}(x) = \sin^{-1}\left(\frac{3}{5}\right) \]
Step 4: Final value
\[ x = \frac{3}{5} \]
Final Answer
\[ \boxed{\frac{3}{5}} \]
Explanation
cotθ = 0 when θ = π/2, then apply identity cos⁻¹x = π/2 − sin⁻¹x.