(sin x + sin 3x + sin 5x )/ (cos x + cos 3x + cos 5x) = tan 3x

Question

how do you prove this identity (using a level maths)

Answer 1

sin 3x = 3 sin x - 4 sin³ x

sin 5x = 5 sin x - 20 sin³ x + 16 sin⁵ x

 

cos 3x = -3 cos x + 4 cos³ x

cos 5x = 5 cos x - 20 cos³ x + 16 cos⁵ x

 

Substitute, combine, and reduce.

Answer 2

sin x + sin 5x = 2sin 3x cos 2x

The numerator then becomes sin 3x (2 cos 2x +1).

 

cos x + cos 5x = 2 cos 3x cos 2x

The denominator then becomes cos 3x (2cos 2x +1)

 

As long as 2 cos 2x + 1 ≠ 0, it divides out (cancels) and then the fraction is just (sin 3x)/cos 3x = tan 3x