Question

How do you prove `cos(3x)= 4cos^3(x)-3cos(x)` using the double angle identity?

Answer 1

It's all about trigonometry (duh)

Explanation:

LHS
`= cos 3x`
`= cos (x + 2x)`
`= cos x cos 2x - sin x sin 2x` [note `cos 2x` and `sin 2x`]
`= cos x (2 cos^2 x - 1) - sin x (2 sin x cos x)` [open the brackets]
`= 2 cos^3 x - cos x - 2 sin^2 x cos x`
`= 2 cos^3 x - cos x - 2 (1 - cos^2 x) cos x` [`sin^2 x + cos^2 x = 1`]
`= 2 cos^3 x - cos x - 2 (cos x - cos^3 x)` [open the brackets]
`= 2 cos^3 x - cos x - 2 cos x + 2 cos^3 x`
`= 4 cos^3 x - 3 cos x` [collect terms]
`=` RHS