Last updated at May 29, 2018 by Teachoo

Misc 5 Prove that: sin x + sin 3x + sin5x + sin 7x = 4 cos x sin 2x sin 4x Solving L.H.S sin x + sin 3x + sin5x + sin 7x = (sin 7x + sin x ) + (sin 5x + sin 3x) = (2 sin ((7x + x)/2) cos((7x x)/2) ) + (2 sin ((5x + 3x)/2) cos ((5x 3x)/2) ) = 2 sin 8 /2 cos 6 /2 + 2 sin 8 /2 cos 2 /2 = 2 sin 4x cos 3x + 2 sin 4x cos x = 2 sin 4x [ cos 3x + cos x] = 2sin 4x [ 2cos ((3x + x)/2) cos ((3x x)/2) ] = 2sin 4x [ 2cos (4x/2) cos (4x/2) ] = 2 sin 4x [2cos 2x . cos x ] = 2 sin 4x . 2 cos 2x . cos x = 4 cos x . cos 2x . sin 4x = R.H.S Hence L.H.S = R.H.S Hence proved