Expert: Sherry Wallin - 8/8/2011

Question
Prove that
sin16x/sinx=16cosx.cos2x.cos4x.cos8x

Answer
Hi Dhananjay~

Sorry for the delay in answering I have been on vacation in the mountains with no internet access...

sin 16x/sin x = sin (8x + 8x)/sin x
= (sin 8x cos 8x + sin 8x cos 8x)/ sin x
= sin 8x( 2cos 8x)/sin x
= 2sin 8x cos 8x/sin x
= 2 sin(4x + 4x)cos 8x/sin x
= 2 sin 4x(2cos 4x)cos 8x/sin x
= 4 sin 4x cos 4x cos 8x/sin x
= 4 sin(2x + 2x)cos 4x cos 8x/sin x
= 4 (sin 2x cos 2x + cos 2x  sin 2x) cos 4x cos 8x/sin x
= 4 sin 2x (2 cos 2x)cos 4x cos 8x/sin x
= 8 sin 2x cos 2x cos 4x cos 8x/sin x
= 8 sin(x + x)cos 2x cos 4x cos 8x/sin x
= 8 (sin x cos x + sin x cos x)cos 2x cos 4x cos 8x/sin x
= 8 sin x(2cos x)cos 2x cos 4x cos 8x/sin x
= 16 sin x(cos x cos 2x cos 4x cos 8x)/sin x
= 16 cos x cos 2x cos 4x cos 8x

Math Prof