To prove this equation, we can start by using the trigonometric identity: sin^2x + cos^2x = 1

First, let's write the left side of the given equation using this identity:

sin^2x - cos^2x = sin^2x - (1 - sin^2x) = 2sin^2x - 1

Now, we need to simplify the right side of the given equation:

√2 sin2x = √2 2sinxcosx = 2(√2)sinxcosx

Since we want to have 2sin^2x on the right side, let's convert cosx to 1 - sin^2x:

2(√2)sinxcosx = 2(√2)sinx(1 - sin^2x) = 2(√2)sinx - 2(√2)sin^3x

Now, we can see that it won't simplify directly to 2sin2x, so it seems the given equation is not correctly stated.

Let me know if you would like me to proceed with the available information, or if you have any other equations or questions to solve.