<s>[INST] prove $2\sin^4(x)+2\cos^4(x)+\sin^2(2x)=2$ \begin{align}2\sin^4(x) + 2\sin^2(x)\cos^2(x) + 2\cos^4(x)&=2(\sin^4(x) + \sin^2(x)\cos^2(x) + \cos^4(x))\\
&=2(\sin^2(x) + \cos^2(x))^2 -\sin^2(x)\cos^2(x)\\
&=2(1) -\sin^2(x)\cos^2(x) \\
&=2-\sin^2(x)\cos^2(x)
\end{align}
And I'm stuck 
 [/INST] $$\sin^2 2x = (\sin 2x)^2 = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x,$$ therefore, the LHS is $$2 \sin^4 x + 4 \sin^2 x \cos^2 x + 2 \cos^4 x,$$ which immediately factors as $$2 (\sin^2 x + \cos^2 x)^2 = 2.$$
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