Question

(1+sin2a + cos2a)^2 = 4 cos^2a(1+ sin2a)

Answer 1

Proof.

It’s necessary to prove that

$\left(1+\sin 2x+\cos 2x\right)^{2}=4\cos^{2}x\left(1+\sin 2x\right).$ (1)

Consider the left side of this equality. It’s well known that $\sin 2x=2\sin x\cos x$ and $\cos 2x=2\cos^{2}x-1$. Having substituted these equalities to (1), we get the following:

$\left(1+\sin 2x+\cos 2x\right)^{2}=\left(1+2\sin x\cos x+2\cos^{2}x-1\right)^{2}=4\cos^{2}x\left(\sin x+\cos x\right)^{2}.$ (2)

As $\sin^{2}x+\cos^{2}x=1$, we have:

$4\cos^{2}x\left(\sin x+\cos x\right)^{2}=4\cos^{2}x\left(\sin^{2}x+\cos^{2}x+2\cos x\sin x\right)=$

$=4\cos^{2}x\left(1+2\cos x\sin x\right)=4\cos^{2}x\left(1+\sin 2x\right).$

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