Problem
Evaluate: \( \cos\left(\sin^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{5}{13}\right)\right) \)
Solution
Let:
\[ A = \sin^{-1}\left(\frac{3}{5}\right), \quad B = \sin^{-1}\left(\frac{5}{13}\right) \]
Step 1: Find cos A and cos B
\[ \sin A = \frac{3}{5} \Rightarrow \cos A = \frac{4}{5} \]
\[ \sin B = \frac{5}{13} \Rightarrow \cos B = \frac{12}{13} \]
Step 2: Use identity
\[ \cos(A+B) = \cos A \cos B – \sin A \sin B \]
Step 3: Substitute
\[ = \frac{4}{5}\cdot\frac{12}{13} – \frac{3}{5}\cdot\frac{5}{13} \]
\[ = \frac{48}{65} – \frac{15}{65} = \frac{33}{65} \]
Final Answer
\[ \boxed{\frac{33}{65}} \]
Explanation
Use triangle method to find cosine values and apply cos(A+B) identity.