How to evaluate $\cos\left(\frac{1}{2}\arcsin\left(\frac{1}{4}\right)\right)$

Question

Question

How to evaluate $\cos(\frac{1}{2} \arcsin(\frac{1}{4}))$?

My thoughts

I tried to do double angle formula as in $\cos(2*\frac{1}{4}\arcsin(\frac{1}{4}))$

But that didn't work at all so then I tried to evaluate the arcsin inside of the expression but I had no success in finding the answer.

Answer 1

$\cos (x) = 2 \cos (x/2)^2-1,$ so $\cos(x/2) = \frac12 \sqrt{1+ \cos(x)}.$ Now, $\cos \arcsin (x) = \sqrt{1-x^2}.$ At this point you should be able to finish.