proof "$cos^2(\frac{x}{2})=\frac{1+cos(x)}{2}$" I want to proof: 
$cos^2(\frac{x}{2})=\frac{1+cos(x)}{2}$
I have changed the given eqation $1 = cos^2(x)+sin^2(x)$
$\to$ $cos^2(x) = 1- sin^2(x)$
Then another given eqation: 
$cos(2x) = cos^2(x)-sin^2(x)$ $\to$ $-sin^2(x)=cos(2x)-cos^2(x)$
After that I have put those equtions together: 
$$cos^2(x)=1-sin^2(x)$$
$$cos^2(x)=1+cos(2x)-cos^2(x)$$
So I have the 1+ {...} structure. How can I go on ?
Or is that the wrong way to proof the equation?
I would really appreciate some hints. 

$$\cos(2y)=2\cos^2(y)-1$$
Let $2y=x$,
$$\cos(x )= 2\cos^2\left(\frac{x}{2}\right)-1$$
Hence 
$$\frac{1+\cos(x)}{2}=\frac{1+2\cos^2(\frac{x}{2})-1}{2}=\cos^2\left(\frac{x}{2}\right)$$
