Question

Can you check and see if I am doing this right

Answer 1

Given:

$\cos ^2\mleft(\frac{1}{2}\cos ^{-1}\mleft(-\frac{4}{5}\mright)\mright)$

To find the exact value:

We know the formula,

$\cos ^2x=\frac{1+\cos 2x}{2}$

If we take,

$x=(\frac{1}{2}\cos ^{-1}(-\frac{4}{5}))$

Then, the given problem can be rewritten as,

$\begin{gathered} \cos ^2(\frac{1}{2}\cos ^{-1}(-\frac{4}{5}))=\frac{1+\cos\lbrack2(\frac{1}{2}\cos^{-1}(-\frac{4}{5}))\rbrack}{2} \\ =\frac{1+\cos \lbrack\frac{2}{2}\cos ^{-1}(-\frac{4}{5})\rbrack}{2} \\ =\frac{1+\cos \lbrack\cos ^{-1}(-\frac{4}{5})\rbrack}{2} \end{gathered}$

Cancelling cos and its inverse we get

$\begin{gathered} =\frac{1+(-\frac{4}{5})}{2} \\ =\frac{1-\frac{4}{5}}{2} \\ =\frac{\frac{1}{5}}{2} \\ =\frac{1}{10} \end{gathered}$

Hence, the exact value is,

$\frac{1}{10}$