<s>[INST] 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. 
 [/INST] Usually the easiest way to proof trigonometric identities is to use complex exponentials. Here we have $\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})$ and thus:
$$ \cos^2(\frac{x}{2})
= \frac{1}{4}(e^{i\frac{x}{2}}+e^{-i\frac{x}{2}})^2
= \frac{1}{4}(e^{ix} + 2+ e^{-ix}) = \frac{1}{2}+\frac{1}{2}\cos(x) $$
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