Question

How do you prove that `2cos^4x + 2sin^2xcos^2x = cos2x + 1`?

Answer 1

Use the identities:
`color(white)("XXX")cos^2(x)+sin^2(x)=1` (Pythagorean)
and
`color(white)("XXX")cos(2x)=2cos^2(x)-1` (Double angle)

Explanation:

`2cos^4(x)+2xin^2(x)cos^2(x)`

`=(2cos^2(x))*(cos^2(x)+sin^2(x))`

`=2cos^2(x)`

`=cos(2x)+1` since by double angle formula: `cos(2x)=2cos^2(x)-1`