Question

Find the exact value of each expression.

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

Short answer

The value of the expression is$\frac{63}{65}$

Step-by-step solution

Step 1. Given information

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

Step 2. Solve the expression

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

So,

$\sin \alpha =\frac{5}{13},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$

$\tan \beta =\frac{3}{4},-\frac{\mathrm{\pi }}{2}\le \beta \le \frac{\mathrm{\pi }}{2}$

$\cos \alpha =\frac{12}{13}$

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

Now,

$$\begin{gathered} \cos \left(\alpha -\beta \right) \\ =\cos \alpha ·\cos \beta +\sin \alpha ·\sin \beta \\ =\frac{12}{13}·\frac{4}{5}+\frac{5}{13}·\frac{3}{5} \\ =\frac{63}{65} \end{gathered}$$