Please show how to solve-

Question

4sin x*sin 2x*sin 4x=sin 3x

Answer 1

Given that 4sinx*sin2x*sin4x=sin3x

LHS= 4sinx*sin2x*sin4x

        = 4sinx*2sinxcosx*sin4x      [ Since sin2x = 2sinxcosx ]

        = 2(2sinx*2sinxcosx)sin4x

        = 2(4sin^2xcosx)sin4x          [ Since cos3x = cosx - 4sin^2xcosx ]

         =2(cosx-cos3x)sin4x

         = 2cosxsin4x - 2cos3xsin4x

         = sin5x+sin3x-sin7x-sinx    [ Where sin5x - sin7x = 0 ]

         = 0 + sin3x -sinx   [ Where sin3x - sinx = 0 => sin3x = sinx ]

         =sin3x

         = RHS

Therefore 4sin x*sin 2x*sin 4x=sin 3x

 

Answer 2

 

Let , the function has a period of , first find all solutions in the interval .

The function is negative in third and fourth quadrant.