Answer
$\frac{3}{5}$
Work Step by Step
Given expression is-
$\cos(\sin^{-1}\frac{4}{5})$
Assuming $\sin^{-1} \frac{4}{5}$ = $u$ , we get-
$\sin u$ = $\frac{4}{5}$, [$-\frac{\pi}{2}\leq u\leq \frac{\pi}{2}$]
Therefore $\cos u$ will be positive.
From first Pythagorean identity-
$\cos u$ = + $\sqrt {1 - \sin^{2} u}$
= $\sqrt {1 - (\frac{4}{5})^{2}} $
= $\sqrt {1 - \frac{16}{25}} $
= $\sqrt { \frac{25 - 16}{25}} $
= $\sqrt { \frac{9}{25}} $ = $\frac{3}{5}$
i.e. $\cos u$ = $\frac{3}{5}$
i.e. $\cos(\sin^{-1}\frac{4}{5})$ = $\frac{3}{5}$
