Question

How do you use the double angle or half angle formulas to prove `cos(3x)= 4cos^3(x)-3cos(x)`?

Answer 1

Prove the trig identity: cos 3a = 4cos^3 x - 3cos x

Explanation:

Apply the trig identity: cos (a + b) = cos a.cos b - sin.a.sin b.
cos 3x = cos (2x + x) = cos 2x.cos x - sin 2x.sin x.
Use the trig identities:
`cos 2x = (2cos ^2 x - 1)`
sin 2x = 2sin x.cos x
`cos 3x = (2cos^2 x - 1)cos x - 2sin^2 x.cos x.`
Put cos x in common factor.
`cos 3x = cos x(2cos^2 x - 1 - 2sin^2 x).`
Replace `(sin^2 x)` by `(1 - cos^2 x)`.
`cos 3x = cos x( 2cos^2 x - 1 - 2 + 2cos^2 x)`
`cos 3x = cos x(4cos^2 x - 3)`
`cos 3x = 4cos^3 x - 3cos x`