verify that the following equation is an identity.\n(2 sin x - 2 cos x)² = -4 sin 2x + 4\nto verify the…

verify that the following equation is an identity.\n(2 sin x - 2 cos x)² = -4 sin 2x + 4\nto 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\n(2 sin x - 2 cos x)²\n= \nsimplify within the parentheses\napply an even - odd identity\napply a cofunction identity\nexpand the expression

verify that the following equation is an identity.\n(2 sin x - 2 cos x)² = -4 sin 2x + 4\nto 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\n(2 sin x - 2 cos x)²\n= \nsimplify within the parentheses\napply an even - odd identity\napply a cofunction identity\nexpand the expression

Answer

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.