(cos 2x + cos 4x) / (cos 2x - cos 4x) = cot 3x / tan x prove the identities?

Question

Show all of your work, including what calculator steps you used, if any.

Answer 1

Since cos(a) + cos(b) = 2cos((a+b)/2) cos((a-b)/2)

and cos(a) - cos(b) = 2sin((a+b)/2) sin((b-a)/2)

Also sin(-a) = -sin(a) and cos(-a) = cos(a)

You can see more Trigonometric Formulas at
http://www.mathcity.org/fsc/trigonometric_formulas.html

Now using above formulas
cos 2x + cos 4x = 2cos(6x/2) cos (-2x/2) = 2cos3x cosx

and (cos 2x - cos 4x) = 2sin(6x/2) sin(2x/2) = 2sin3x sinx

Now finally
(cos 2x + cos 4x) / (cos 2x - cos 4x)
= 2cos3x cosx / 2sin3x sinx
= (cos3x/sin3x) / (sinx/cosx)
= cot3x / tanx

Answer 2

2cos((2x+4x)/2)cos((2x-4x)/2)
/
-2sin((2x+4x)/2)sin((2x-4x)/2))
=cos(3x)cos(x) / -sin3x sin (-x)
=cot3x cotx
=cot3x .(1 / tanx)
=cot3x / tanx