Another math question?

Question

cos(2x)=1-2sin²(x). can anyone demonstrate this equation by using this one: sin(2x)=2sin(x)cos(x)? I'd be so grateful!

Answer 1

cos 2x = cos (x + x)
= cos x.cos x - sin x.sin x
= (1 - sin² x) - sin² x
= 1 - 2 sin² x
sin 2x = sin (x + x)
= sin x cos x + cos x sin x
= 2 sin x cos x

Answer 2

If you are familiar with calculus, you could differentiate both sides of
sin(2x) = 2sin(x)cos(x) to get

2cos(2x) = 2cos(x)cos(x) + 2sin(x){-sin(x)} which simplifies as:
cos(2x) = cos² x - sin² x

Now use cos² x + sin² x = 1 rearranged as
cos²x = 1 - sin² x to eliminate the cos² x.

This gives cos(2x) = 1 - 2sin² x as required.