Question

Find the exact value of each expression.

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

Short answer

The value of the expression is$-1$

Step-by-step solution

Step 1. Given information

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

Step 2. Solve the expression

Let $\alpha ={\sin }^{-1}\left(-\frac{4}{5}\right)$and $\beta ={\tan }^{-1}\frac{3}{4}$

So,

role="math" localid="1646553906966" $\sin \alpha =-\frac{4}{5},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$

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

$$\begin{gathered} \cos \alpha =\sqrt{1-{\sin }^2\alpha } \\ \cos \alpha =\sqrt{1-{\left(-\frac{4}{5}\right)}^2} \\ \cos \alpha =\frac{3}{5} \end{gathered}$$

$$\begin{gathered} se{c}^2\beta =1+{\tan }^2\beta \\ se{c}^2\beta =1+{\left(\frac{3}{4}\right)}^2 \\ se{c}^2\beta ==\frac{25}{16} \\ sec\beta =\frac{5}{4} \\ \cos \beta =\frac{4}{5} \end{gathered}$$

$$\begin{gathered} \sin \beta =\sqrt{1-{\cos }^2\beta } \\ \sin \beta =\sqrt{1-{\left(\frac{4}{5}\right)}^2} \\ \sin \beta =\frac{3}{5} \end{gathered}$$

Now,

$$\begin{gathered} \sin \left(\alpha -\beta \right) \\ =\sin \alpha ·\cos \beta -\cos \alpha ·\sin \beta \\ =-\frac{4}{5}·\frac{4}{5}-\frac{3}{5}·\frac{3}{5} \\ =-1 \end{gathered}$$