Question

Prove the following:

`4 cos x. cos(x + pi/3) . cos (x - pi/3)` = cos 3x

Answer 1

`4 cos x. cos(x + pi/3). cos (x - pi/3)`

Cosine Addition Formula: cos (a+b) = cosa cosb − sina sinb

Cosine Subtraction Formula: cos (a−b) = cosa cosb + sina sinb

Here, a = x and b = `pi/3`

L.H.S. = `4 cos x. cos(x + pi/3). cos (x - pi/3)`

= `4 cosx. (cos x  cos  pi/3 - sin x  sin  pi/3) (cosx  cos  pi/3 + sinx  sin  pi/3)`

= `4 cosx  (cosx  1/2 - sinx  sqrt3/2) (cosx  1/2 + sinx  sqrt3/2)`          ...`[∵ sin  pi/3 = sqrt3/2, cos  pi/3 = 1/2]`

= `4 cos (1/2 cosx - sqrt3/2  sinx) (1/2  cosx + sqrt3/2  sinx)`

= `4 cosx[(1/2 cosx)^2 - (sqrt3/2 sin x)^2]`

= `4 cosx (1/4cos^2x - 3/4 sin^2)`

= `4/4(cos^3x - 3 sin^2.cosx)`

= `cos^3x - 3 cosx.sin^2x`

= cos³x - 3 cosx (1 - cos²x)

= cos³x - 3 cosx + 3 cos³x

= 4 cos³x - 3 cosx

= cos 3x

= R.H.S.