i need help on this identity, cos^2x - sin^2x = 2cos^2x-1?

Question

Identity sucks, i need help step by step would help thanks

Answer 1

You need to know cos^2 x + sin^2 x = 1. We can also write this as sin^2 x = 1 - cos^2 x. From there, it's simple: substitute 1 - cos^2 x for sin^2 x in your equation to get
cos^2 x - (1 - cos^2 x)
= cos^2 x - 1 + cos^2 x
= 2 cos^2 x - 1.

The main interest of this particular formula is that cos^2 x - sin^2 x and 2cos^2 x - 1 are equivalent ways of expressing cos (2x). (A useful exercise for you would be to prove that 1 - 2 sin^2 x is also equivalent.)

Answer 2

The negative 1 on the rightmost side of the equal sign has the following identity. From the Pythagorean theorem.

sin^2x+cos^2x= 1

So we now have

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

releasing from the brackets yields the following:

Left side = 2cos^2x - sin^2x - cos^2x

The -sin^2x on both sides of the equal sign cancel off.
This leaves:
cos^2x = 2cos^2 x - cos^2 x

adding cos^2x to both sides gives us :

2cos^2x = 2cos^2x

use of brackets omitted for expediency
hope this helps

Answer 3

sin^2 = 1 - cos^2 Substituting this into your expression we get

cos^2x - 1 + cos^2 which becomes
2cos^2x -1