Find the exact value of the expression. \[ \cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{5}{12}\right) \] \( \cos \left(\sin ^{-1} \frac{4}{5}+\tan ^{-1} \frac{5}{12}\right)=\square \) (Type an exact answer in simplified form.)

To find the exact value of the expression \( \cos \left(\sin^{-1} \frac{4}{5} + \tan^{-1} \frac{5}{12}\right) \), we will use the cosine addition formula, which states that \[ \cos(a + b) = \cos a \cos b - \sin a \sin b. \] Here, let \( a = \sin^{-1} \frac{4}{5} \) and \( b = \tan^{-1} \frac{5}{12} \). ### Step 1: Calculate \( \cos a \) and \( \sin a \) From \( a = \sin^{-1} \frac{4}{5} \): - We have \( \sin a = \frac{4}{5} \). - To find \( \cos a \), we use the Pythagorean identity \( \cos^2 a + \sin^2 a = 1 \): \[ \cos^2 a = 1 - \sin^2 a = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}. \] Thus, \[ \cos a = \sqrt{\frac{9}{25}} = \frac{3}{5}. \] ### Step 2: Calculate \( \cos b \) and \( \sin b \) From \( b = \tan^{-1} \frac{5}{12} \): - We can interpret this as a right triangle where the opposite side is \( 5 \) and the adjacent side is \( 12 \). Using the Pythagorean theorem, the hypotenuse \( h \) is: \[ h = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13. \] - Therefore, we have: - \( \sin b = \frac{5}{13} \) - \( \cos b = \frac{12}{13} \) ### Step 3: Combine to find \( \cos(a + b) \) Now substituting into the cosine addition formula: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b. \] Substituting the values: \[ \cos(a + b) = \left(\frac{3}{5}\right)\left(\frac{12}{13}\right) - \left(\frac{4}{5}\right)\left(\frac{5}{13}\right). \] Calculating each term: 1. \( \frac{3}{5} \cdot \frac{12}{13} = \frac{36}{65} \) 2. \( \frac{4}{5} \cdot \frac{5}{13} = \frac{20}{65} \) Now, substituting back into the equation gives: \[ \cos(a + b) = \frac{36}{65} - \frac{20}{65} = \frac{16}{65}. \] Thus, the exact value of the expression is: \[ \boxed{\frac{16}{65}}. \]