How to change cos4x into 2cos^2 2x-1?

cos4x

cos(2x + 2x)

cos2xcos2x - sin2xsin2x

cos^2 2x - sin^2 2x

As sin^2 2x = 1 - cox^2 2x, So

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

cos^2 2x - 1 + cos^2 2x

cos^2 2x + cos^2 2x - 1

2cos^2 2x - 1

Hence

cos4x = 2cos^2 2x - 1

How do you get sin^2 2x = 1 - cos^2 2x?

Because its one of the basic identity. which is

sin^2 x + cos^2 x = 1

So in the question, there is 2x in the angle, so we put 2x

sin^2 2x + cos^2 2x = 1

Now solving for sin^2 2x, we get

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

Okay.

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