Chapter 9.5, Problem 46E

To determine

To find: thevalue of $\cos \left(\alpha +\beta \right)$ .

Answer to Problem 46E

The value is $\cos \left(\alpha +\beta \right)=\frac{16}{65}$ .

Explanation of Solution

Given information:

The values are $\tan \alpha =\frac{4}{3}$ and $\cot \beta =\frac{5}{12}$ .

Calculation: since, $\tan \alpha =\frac{4}{3}$ ,so

  $\cos \alpha =\frac{3}{5}\text{ and sin}\alpha =\frac{4}{5}$

Since, $\cot \beta =\frac{5}{12}$ so

  $\cos \beta =\frac{12}{13}\text{ and sin}\beta =\frac{5}{13}$

Now, to find the value of $\cos \left(\alpha +\beta \right)$ , use the formula and then substitute and simplify,

  $\begin{array}{l}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ \cos \left(\alpha +\beta \right)=\frac{3}{5}\left(\frac{12}{13}\right)-\frac{4}{5}\left(\frac{5}{13}\right)\text{ [substitute]}\\ \cos \left(\alpha +\beta \right)=\frac{36}{65}-\frac{20}{65}\text{ [multiply]}\\ \cos \left(\alpha +\beta \right)=\frac{16}{65}\text{ [subtract]}\end{array}$

Thus, the value is $\cos \left(\alpha +\beta \right)=\frac{16}{65}$ .