<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] Let there is a function $f(x) = \cos x + 1$, then $$f^2(x) = \cos^2 x + 2\cos x + 1 = 2(1 + \cos x) - \sin^2 x.$$ While $$f(x) = 2\cos^2\frac{x}{2} = \sqrt{4\cos^4\frac{x}{2}}.$$ Therefore $$ 4\cos^4\frac{x}{2} = 2(1 +\cos x) - \sin^2x.$$
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