Question

In Problems 79-90, find the exact value of each expression.

${\cos }^2\left(\frac{1}{2}{\sin }^{-1}\frac{3}{5}\right)$

${\cos }^2\left(\frac{1}{2}{\sin }^{-1}\frac{3}{5}\right)$

Short answer

The exact value is$\frac{9}{10}$

Step-by-step solution

Given information

Given expression${\cos }^2\left(\frac{1}{2}{\sin }^{-1}\frac{3}{5}\right)$

Use half angle formula and calculate

Here,

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

Calculating using half angle formula

Calculating, we get

$$\begin{gathered} a={\cos }^2\left(\frac{1}{2}{\sin }^{-1}\left(\frac{3}{5}\right)\right) \\ a=\frac{1+\cos \left({\cos }^{-1}\left(\frac{4}{5}\right)\right)}{2} \\ a==\frac{1+\frac{4}{5}}{2} \\ a=\frac{9}{10} \end{gathered}$$