Answer
$\frac{\sqrt 5}{2}$
Work Step by Step
Given expression is-
$\cot(\sin^{-1}\frac{2}{3})$
Assuming $\sin^{-1} \frac{2}{3}$ = $u$ , we get-
$\sin u$ = $\frac{2}{3}$, [$-\frac{\pi}{2}\leq u\leq \frac{\pi}{2}$]
From First Pythagorean identity-
$\cos u$ = + $\sqrt {1 - \sin^{2} u}$
= $\sqrt {1 - (\frac{2}{3})^{2}} $
= $\sqrt {1 - \frac{4}{9}} $
= $\sqrt { \frac{9 - 4}{9}} $
= $\sqrt { \frac{5}{9}} $ = $\frac{\sqrt 5}{3}$
i.e. $\cos u$ = $\frac{\sqrt 5}{3}$
We know that, $\cot u$ = $\frac{\cos u}{\sin u}$ = $\frac{\frac{\sqrt 5}{3}}{\frac{2}{3}}$ = $\frac{\sqrt 5\times3}{3\times 2}$
i.e. $\cot u$ = $\frac{\sqrt 5}{2}$
i.e. $\cot(\sin^{-1}\frac{2}{3})$ = $\frac{\sqrt 5}{2}$
