Question

Question 5 Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x

Class X1 - Maths -Trigonometric Functions Page 82

Answer 1

LHS =sinx + sin3x + sin5x + sin7x
= (sinx + sin3x) + (sin5x + sin7x)
use the formula ,

sinC + sinD = 2sin(C+D)/2.cos(C-D)/2

= 2sin(x + 3x)/2.cos(3x -x)/2 + 2sin(5x+7x)/2.cos(7x-5x)/2

= 2sin2x.cosx + 2sin6x.cosx
= 2cosx [sin2x + sin6x ]
= 2cosx [ 2sin(2x + 6x)/2.cos(6x-2x)/2]
=2cosx. [ 2sin4x.cos2x ]
= 4cosx.cos2x.sin4x = RHS