Evaluate: `cos(sin^(-1)(3/5)+sin^(-1)(5/(13)))`

Explanation

To evaluate the expression \( \cos(\sin^{-1}(\frac{3}{5}) + \sin^{-1}(\frac{5}{13})) \), we can use the formula for the cosine of the sum of two angles. The formula states:

\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

where \( A = \sin^{-1}(\frac{3}{5}) \) and \( B = \sin^{-1}(\frac{5}{13}) \).

### Step 1: Find \( \cos A \) and \( \sin A \)

1. Since \( A = \sin^{-1}(\frac{3}{5}) \), we have:
\[ \sin A = \frac{3}{5} \]
To find \( \cos A \), we use the identity \( \cos^2 A + \sin^2 A = 1 \):
\[ \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos A = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

### Step 2: Find \( \cos B \) and \( \sin B \)

2. Since \( B = \sin^{-1}(\frac{5}{13}) \), we have:
\[ \sin B = \frac{5}{13} \]
To find \( \cos B \), we again use the identity:
\[ \cos^2 B = 1 - \sin^2 B = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} \]
Thus,
\[ \cos B = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

### Step 3: Substitute into the cosine sum formula

3. Now substitute \( \sin A \), \( \cos A \), \( \sin B \), and \( \cos B \) into the cosine sum formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Substituting the values we found:
\[ \cos(A + B) = \left(\frac{4}{5}\right)\left(\frac{12}{13}\right) - \left(\frac{3}{5}\right)\left(\frac{5}{13}\right) \]

### Step 4: Calculate the expression

4. Calculate each term:
\[ \cos(A + B) = \frac{48}{65} - \frac{15}{65} = \frac{33}{65} \]

### Final Answer

Thus, the value of \( \cos(\sin^{-1}(\frac{3}{5}) + \sin^{-1}(\frac{5}{13})) \) is:
\[ \boxed{\frac{33}{65}} \]