<s>[INST] Simplifying $ 2\cos^2(x)\sin^2(x) + \cos^4(x) + \sin^4(x)$ who can simplify the following term in a simplest way?
$$ 2\cos^2(x)\sin^2(x) + \cos^4(x) + \sin^4(x)$$
(The answer is 1). Thanks for any suggestions.
 [/INST] Although the answer is straight forward but here are some steps if you find helpful
Notice, $$2\cos^2 x\sin^2 x+\cos^4 x+\sin^4 x$$ $$=(\cos^2 x)^2+(\sin^2 x)^2+2(\cos^2 x)(\sin^2 x)$$ Now, assume $\alpha =\cos^2 x$ & $\beta=\sin^2 x$ & apply $\alpha^2+\beta^2+2\alpha \beta=(\alpha+\beta)^2$$$=(\cos^2x+\sin^2 x)^2$$$$=(1)^2=1 \quad (\text{since,}\ \cos^2 x+\sin^2 x=1)$$
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