<s>[INST] trigonometry identity I have some problem with proving this identity: 
$$2\left(1+\cos\alpha \right)-\sin^2\alpha=4\cos^4\frac{\alpha}{2}$$
I tried to start from the right side rewritting it to $(2\cos^2(2\frac{\alpha}{4}))^2$ but it's not working. 
 [/INST] We have the double angle identities $sin(x)=2sin(\frac{x}{2})cos(\frac{x}{2})$ and $cos(x)=2cos^2(\frac{x}{2})-1$
Substituting into the left hand side of your expression, we have 
$2(1+2cos^2(\frac{\alpha}{2})-1) - (2sin(\frac{x}{2})cos(\frac{x}{2}))^2$
$=4cos^2(\frac{\alpha}{2})-4sin^2(\frac{\alpha}{2})cos^2(\frac{\alpha}{2})$
$=4cos^2(\frac{\alpha}{2})[1-sin^2(\frac{\alpha}{2})]$
$=4cos^2(\frac{\alpha}{2})[cos^2(\frac{\alpha}{2})]$
$=4cos^4(\frac{\alpha}{2})$
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