Question

If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)

Answer 1

Given: 
\[ \sin A = \frac{4}{5}\text{ and }\cos B = \frac{5}{13}\]
We know that
\[ \cos A = \sqrt{1 - \sin^2 A}\text{ and }\sin B = \sqrt{1 - \cos^2 B} ,\text{ where }0 < A , B < \frac{\pi}{2}\]
\[ \Rightarrow \cos A = \sqrt{1 - \left( \frac{4}{5} \right)^2} \text{ and }\sin B = \sqrt{1 - \left( \frac{5}{13} \right)^2}\]
\[ \Rightarrow \cos A = \sqrt{1 - \frac{16}{25}}\text{ and }\sin B = \sqrt{1 - \frac{25}{169}}\]
\[ \Rightarrow \cos A = \sqrt{\frac{9}{25}}\text{ and }\sin B = \sqrt{\frac{144}{169}}\]
\[ \Rightarrow \cos A = \frac{3}{5}\text{ and }\sin B = \frac{12}{13}\]
Now,
\[ \cos\left( A - B \right) = \cos A \cos B + \sin A \sin B\]
\[ = \frac{3}{5} \times \frac{5}{13} + \frac{4}{5} \times \frac{12}{13}\]
\[ = \frac{15}{65} + \frac{48}{65}\]
\[ = \frac{63}{65}\]