How to derive this equation in many ways?

Question

Given the equation: $$y=\sin^4(x)\cdot\cos^4(x)$$ I have derived the equation using product rule: $$f'(x)\cdot g(x)+f(x)\cdot g'(x)$$ Then I`ve got $$y'=4\sin^3(x)\cos^5(x)-4\sin^5(x)\cos^3(x)$$, then factoring my final is $$y'=4\sin^3(x)\cos^3(x)[1-2\sin^2(x)]$$. How about using identities before deriving? Does it give the same answer since there are many identities we can substitute from.

Answer 1

HINT:

Use $\sin2A=2\sin A\cos A,\cos2B=2\cos^2B-1=1-2\sin^2B, $

$$16y=(2\sin x\cos x)^4=\sin^42x$$

$$64y=(2\sin^22x)^2=(1-\cos4x)^2=1-2\cos4x+\cos^24x$$

$$128y=2-4\cos4x+(1+\cos8x)$$