Question

Write the value of cos² $\left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)$.

Answer 1

$$\begin{gathered} \mathrm{Let}y={\cos }^{-1}\left(\frac{3}{5}\right) \\ \Rightarrow \cos y=\frac{3}{5} \end{gathered}$$
Now,
$$\begin{gathered} {\cos }^2\left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)={\cos }^2\left(\frac{1}{2}y\right) \\ =\frac{\cos y+1}{2}\left[\because \cos 2x=2{\cos }^{2}x-1\right] \\ =\frac{\frac{3}{5}+1}{2} \\ =\frac{\frac{8}{5}}{2} \\ =\frac{4}{5} \end{gathered}$$
∴ ${\cos }^2\left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)=\frac{4}{5}$