Questions: Give the exact value of the expression without using a calculator. cos(tan^(-1)(-12/5) + tan^(-1)(4/3)) cos(tan^(-1)(-12/5) + tan^(-1)(4/3))=□ (Simplify your answer, including any radicals. Use integers or fractions for any numbers)

Solution Steps

Given \( A = \tan^{-1}\left(-\frac{12}{5}\right) \), we can find \( \cos A \) and \( \sin A \) using the definitions of cosine and sine in a right triangle. The values are calculated as follows: \[ \cos A = \frac{1}{\sqrt{1 + \left(-\frac{12}{5}\right)^2}} = \frac{1}{\sqrt{1 + \frac{144}{25}}} = \frac{1}{\sqrt{\frac{169}{25}}} = \frac{5}{13} \] \[ \sin A = \frac{-\frac{12}{5}}{\sqrt{1 + \left(-\frac{12}{5}\right)^2}} = \frac{-\frac{12}{5}}{\sqrt{\frac{169}{25}}} = \frac{-12}{13} \]

For \( B = \tan^{-1}\left(\frac{4}{3}\right) \), we find \( \cos B \) and \( \sin B \) similarly: \[ \cos B = \frac{1}{\sqrt{1 + \left(\frac{4}{3}\right)^2}} = \frac{1}{\sqrt{1 + \frac{16}{9}}} = \frac{1}{\sqrt{\frac{25}{9}}} = \frac{3}{5} \] \[ \sin B = \frac{\frac{4}{3}}{\sqrt{1 + \left(\frac{4}{3}\right)^2}} = \frac{\frac{4}{3}}{\sqrt{\frac{25}{9}}} = \frac{4}{5} \]

Now, we apply the cosine angle addition formula: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] Substituting the values we calculated: \[ \cos(A + B) = \left(\frac{5}{13}\right) \left(\frac{3}{5}\right) - \left(-\frac{12}{13}\right) \left(\frac{4}{5}\right) \] Calculating this gives: \[ \cos(A + B) = \frac{15}{65} + \frac{48}{65} = \frac{63}{65} \]

Final Answer