Question

Find the exact value of each expression.

$\cos \left({\tan }^{-1}\frac{5}{12}-{\sin }^{-1}\left(-\frac{3}{5}\right)\right)$

Short answer

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

Step-by-step solution

Step 1. Given information

The given expression is$\cos \left({\tan }^{-1}\frac{5}{12}-{\sin }^{-1}\left(-\frac{3}{5}\right)\right)$

Step 2. Solve the expression

Let $\alpha ={\tan }^{-1}\frac{5}{12}$and $\beta ={\sin }^{-1}\left(-\frac{3}{5}\right)$

So,

$\tan \alpha =\frac{5}{12},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$and $\sin \beta =-\frac{3}{5},-\frac{\mathrm{\pi }}{2}\le \beta \le \frac{\mathrm{\pi }}{2}$

$\cos \beta =\frac{4}{5}$

$$\begin{gathered} sec\alpha =\sqrt{1+{\tan }^2\alpha } \\ sec\alpha =\sqrt{1+{\left(\frac{5}{12}\right)}^2} \\ sec\alpha =\frac{13}{12} \\ \cos \alpha =\frac{12}{13} \\ \sin \alpha =\frac{5}{13} \end{gathered}$$

Now,

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