Question

Find the exact value of cos^2(1/2cos^-1 5/13)

Answer 1

Let \(\large \cos^{-1}\frac{5}{13} = x\). Then \(\large \cos(x) = \frac{5}{13}\)
\[
\cos^2(1/2\cos^{-1} 5/13) = \cos^2(x/2) \\
\cos(2A) = 2\cos^2A - 1 \\
2\cos^2(A) = \cos(2A) + 1 \\
\cos^2(A) = \frac 12 * (\cos(2A) + 1)\\
\cos^2(x/2) = \frac 12 * (\cos(x) + 1) = \frac 12 * \left ( \frac {5}{13}+1\right ) =
\frac 12 * \frac {18}{13} = \frac{9}{13}

\]