calculus - Trigonometry Identity (Proof): $sin^4theta +cos^4 theta =1-2sin^2 theta cos^2 theta$

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?