Question

sin 7x - sin 5x /cos 7x + cos 5x= tan x​

Answer 1

Answer:

If you don’t know the formulas used by Azmat Ali in his answer but do know the ‘sum of angle formulas’, you can reason as follows:

cos7x = cos(6x+x) = cos6xcosx - sin6xsinx

cos5x = cos(6x-x) = cos6xcosx + sin6xsinx

hence cos5x + cos7x = 2cos6xcosx (1)

sin7x = sin(6x+x) = sin6xcosx + cos6xsinx

sin5x = sin(6x-x) = sin6xcosx - cos6xsinx

hence sin5x + sin7x = 2cos6xsinx (2)

Now divide (1) by (2).

Step-by-step explanation:

Formulas:

cos x + cos y = 2 cos ((x + y) / 2) cos ((x - y) / 2)

sin x - sin y = 2 sin((x - y) / 2) cos ((x + y) / 2)

So cos 7x + cos 5x = 2 cos 6x cos x and sin 7x - sin 5x = 2 sin x cos 6x

(cos 7x + cos 5x) / (sin 7x - sin 5x) = 2 cos 6x cos x / 2 sin x cos 6x = cos x / sin x = cot x