<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] $$ \sin^2 2x = 4\sin^2 x \cos^2 x$$
Let $\sin x = a$ and $\cos x = b$
RHS of your equation becomes:
$$ 2a^4 + 4a^2b^2+2b^4 = 2(a^2+b^2)^2= 2(\sin^2 x + \cos^2 x)^2 = 2 =RHS$$
</s>