Calc problem, prove this please?

Question

prove sin4x/sinx = 4cosxcos2x

show all work please.

your help is greatly appreciated.

Answer 1

Hello

We can prove this quickly with deMoivre's Theorem.

recall: (cos x + i sin x)^n = cos nx + i sin nx
real and imaginary components on each side must match.

Therefore, sin(4x) = Im [(cis x)^4].

Expanding the (cos x + i sin x)^4 and equating the imaginary components, you have

sin(4x) = 4 cos^3 x sin x - 4 sin^3 cos x
= 4 cos x sin x (cos^2 x - sin^2 x)
= 4 cos x sin x cos 2x

finally divide by sin x to get the desired equality.

Answer 2

sin 4x= sin(2x+2x) =2sin 2x*cos2x and sin 2x = 2sin x*cos x
so sin4x/sinx= 4cosx cos2x and nothing else