<s>[INST] How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$ Prove the following equation.

\begin{eqnarray}
\\\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1\\
\end{eqnarray}

I can't prove it by many methods I use.
Please give me some hints.

Thank you for your attention
 [/INST] It seems one sign in your equality is wrong. One actually has
\begin{align}
1=\left(\cos^2x+\sin^2x\right)^3&=\cos^6x+3\cos^4x\sin^2x+3\cos^2x\sin^4x+\sin^6x=\\
&=\cos^6x+3\sin^2x\cos^2x\underbrace{\left(\cos^2x+\sin^2x\right)}_{=1}+\sin^6x=\\
&=\cos^6x+3\sin^2x\cos^2x+\sin^6x.
\end{align}
</s>