Question

Find the exact value of each expression.

$\cos \left({\tan }^{-1}\frac{4}{3}+{\cos }^{-1}\frac{5}{13}\right)$

Short answer

The value of the expression is$-\frac{33}{65}$

Step-by-step solution

Step 1. Given information

The given expression is$\cos \left({\tan }^{-1}\frac{4}{3}+{\cos }^{-1}\frac{5}{13}\right)$

Step 2. Solve the expression

Let $\alpha ={\tan }^{-1}\frac{4}{3}$and $\beta ={\cos }^{-1}\frac{5}{13}$

So,

localid="1646557697119" role="math" $\tan \alpha =\frac{4}{3},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$and localid="1646557725031" $\cos \beta =\frac{5}{13},0\le \beta \le \le \mathrm{\pi }$

$$\begin{gathered} \sin \alpha =\frac{4}{5} \\ \cos \alpha =\frac{3}{5} \end{gathered}$$

$$\begin{gathered} \sin \beta =\sqrt{1-{\left(\frac{5}{13}\right)}^2} \\ \sin \beta =\frac{12}{13} \end{gathered}$$

Now,

localid="1646557850672" role="math" $$\begin{gathered} \cos \left(\alpha +\beta \right) \\ =\cos \alpha ·\cos \beta -\sin \alpha ·\sin \beta \\ =\frac{3}{5}·\frac{5}{13}-\frac{4}{5}·\frac{12}{13} \\ =-\frac{33}{65} \end{gathered}$$