Playing With Trig Identities?

Question

(cos(x) - cos5(x))/(sin(x) + sin5(x)) = tan2(x)

How do I handle trig identities that use numbers greater than 2? The above question has a 5? See my point?

Answer 1

cos(A+B) = cosAcosB - sinAsinB
cos(A-B) = cosAcosB + sinAsinB

cos(A+B) + cos(A-B) = 2cosAcosB
cos(A-B) - cos(A+B) = 2sinAsinB
-------------------------

sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB - cosAsinB

sin(A+B) + sin(A-B) = 2sinAcosB
sin(A+B) - sin(A-B) = 2cosAsinB
________________________

(cos(x) - cos(5x))
= cos(3x-2x) - cos(3x+2x)
= 2sin(3x)sin(2x)

sin(x) + sin(5x)
= sin(3x-2x) + sin(3x+2x)
= 2sin(3x)cos(2x)

LHS
= (cos(x) - cos(5x))/(sin(x) + sin(5x))
= 2sin(3x)sin(2x) / (2sin(3x)cos(2x))
= sin(2x)/cos(2x)
= tan(2x)
= RHS

Answer 2

Hint:
remember the formula
cosp-cosq=2sin(p+q)/2 sin(p-q)/2
cosx-cos5x=2sin3x sin (-2x)

do the same for sinx+sin5x