Question

verify that the following equation is an identity.
(2 sin x - 2 cos x)² = -4 sin 2x + 4
to verify the identity, start with the more complicated side and transform it to look like the other side. choose the correct transformation and transform the expression at each step
(2 sin x - 2 cos x)²
=
simplify within the parentheses
apply an even - odd identity
apply a cofunction identity
expand the expression

Explanation:

Step1: Expand the expression

$$(2\sin x - 2\cos x)^2=(2\sin x)^2-2\times(2\sin x)\times(2\cos x)+(2\cos x)^2$$
$$=4\sin^{2}x - 8\sin x\cos x+4\cos^{2}x$$

Step2: Use the double - angle formula and Pythagorean identity

We know that $\sin2x = 2\sin x\cos x$ and $\sin^{2}x+\cos^{2}x = 1$.
So, $4\sin^{2}x + 4\cos^{2}x=4(\sin^{2}x+\cos^{2}x)=4\times1 = 4$, and $-8\sin x\cos x=-4\times(2\sin x\cos x)=-4\sin2x$.
Then $4\sin^{2}x - 8\sin x\cos x+4\cos^{2}x=-4\sin2x + 4$.

Answer:

The left - hand side $(2\sin x - 2\cos x)^2$ simplifies to $-4\sin2x + 4$, which is the right - hand side. So, the equation $(2\sin x - 2\cos x)^2=-4\sin2x + 4$ is an identity.