prove; sin5x + sinx - 2 sin3x ÷cos5x-cosx= tanx
Answers
Answer:
Step-by-step explanation:
L.H.S:
sin 5x + sinx - 2 sin 3x / cos 5x - cos x
= [(sin 5x + sin x) - 2 sin 3x] / (cos 5x - cos x)
//now solve the numerator and denominator separately.
Numerator:
using formula sin x + sin y = 2*sin (x+y/2) * cos(x-y/2)
sin 5x + sin x = 2 * sin(5x+x/2) * cos (5x-x/2)
= 2 * sin (6x/2) * cos (4x/2)
= 2 sin 3x cox 2x
Denominator:
using formula cos x - cos y = - 2*sin (x+y/2) * sin(x-y/2)
cos 5x - cos x = - 2 * sin (5x+x/2) * sin (5x-x/2)
= - 2 * sin (6x/2) * sin (4x/2)
= - 2 sin 3x sin 2x.
//substitute the values back in main equation:
(sin 5x + sin x) - 2 sin 3x / (cos 5x - cos x)
= 2 sin 3x cos 2x - 2 sin 3x / ( -2 sin 3x sin 2x)
= 2 sin 3x (cos 2x - 1) / (-2 sin3x sin 2x)
= (cos 2x -1 ) / - 2 sin 2x
= - (cos 2x - 1) / sin 2x
= 1 - cos 2x / sin 2x
//using formula cos 2x = 1 - 2 sin² x and sin 2x = 2 sin x cos x
= 1 - (1 - 2 sin² x) / 2 sin x cos x
= 2 sin²x / 2 sin x cos x
= sin x / cos x
= Tan x
= R.H.S.
Hence proved.