simplification of $\cos\left( \frac{1}{2}\arcsin (x)\right)$ I know this identity :Simplifying $\cos(\arcsin x)$? but how I can simplify 

$$\cos\left( \frac{1}{2}\arcsin (x)\right)$$

if you have any idea. It could also be a sinus instead of a cosinus. When I do it I loop on something...

Let $c=\cos\left(\frac12\arcsin x\right)$ and let $s=\sin\left(\frac12\arcsin x\right)$. Then  $2cs=\sin(\arcsin x)=x$ and $c^2+s^2=1$. Can you take it from here?
