Problem
Prove: \( \cos^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \cos^{-1}\left(\frac{33}{65}\right) \)
Solution
Let:
\[ A = \cos^{-1}\left(\frac{4}{5}\right), \quad B = \cos^{-1}\left(\frac{12}{13}\right) \]
Step 1: Find sin A and sin B
\[ \cos A = \frac{4}{5} \Rightarrow \sin A = \frac{3}{5} \]
\[ \cos B = \frac{12}{13} \Rightarrow \sin B = \frac{5}{13} \]
Step 2: Use cos(A+B)
\[ \cos(A+B) = \cos A \cos B – \sin A \sin B \]
\[ = \frac{4}{5}\cdot\frac{12}{13} – \frac{3}{5}\cdot\frac{5}{13} \]
\[ = \frac{48}{65} – \frac{15}{65} = \frac{33}{65} \]
Step 3: Conclude
Since \( A, B \in [0, \pi] \), their sum is valid.
\[ A + B = \cos^{-1}\left(\frac{33}{65}\right) \]
Final Result
\[ \boxed{\cos^{-1}\left(\frac{33}{65}\right)} \]
Explanation
Using cosine addition identity and triangle values.