Question

How do you simplify `(cos4x-cos2x)/(sin 4x + sin 2x)`?

Answer 1

`color(crimson)(=> - tan x`

Explanation:

`(cos 4x - cos 2x) / (sin 4x + sin 2x)`

`cos C -cos D = -2 sin (C+D)/2 sin (C-D)/2`

`sin C +sin D = 2 sin (C +D)/2 cos (C -D)/2`

`Nr = -2 sin ((4x+2x)/2) * sin ((4x-2x)/2)`

`Nr = -2 sin 3x * sin x`

`Dr = 2 sin ((4x+2x)/2) * cos((4x-2x)/2)`

`Dr = 2 sin3x * cos x`

`(Nr) / (Dr) =- (cancel2 cancel(sin 3x) * sin x) / (cancel2 cancel(sin 3x) * cos x)`

`color(crimson)(=> - sin x / cos x = - tan x`