Question: Prove that $$ \sin^4\theta +\cos^4 \theta =1-2\sin^2 \theta \cos^2 \theta $$
What I have attempted (Usually I start of with the complex side)
So starting with the LHS
$$ \sin^4\theta +\cos^4 \theta =1-2\sin^2 \theta \cos^2 \theta $$
$$ (\sin^2\theta)^2 + \cos^4 \theta =1-2\sin^2 \theta \cos^2 \theta $$
$$ (1-\cos^2\theta)^2 + \cos^4 \theta =1-2\sin^2 \theta \cos^2 \theta $$
$$ (1-\cos^2\theta)(1-\cos^2\theta) + \cos^4 \theta =1-2\sin^2 \theta \cos^2 \theta $$
$$ 1 - 2\cos^2\theta + \cos^4\theta + \cos^4\theta =1-2\sin^2 \theta \cos^2 \theta $$
$$ 1 - 2\cos^2\theta + 2\cos^4\theta =1-2\sin^2 \theta \cos^2 \theta $$
Now I am stuck... is my approach correct?
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