If sin4 x A B cos 2x C cos 4x then A B C SolutionSolut

If sin^4 x = A + B cos 2x + C cos 4x, then A = B = C =

Solution

Solution:

Given sin^4(x) = A + Bcos(2x) + Ccos(4x)

sin^2 = (1 - cos 2) / 2
cos^2 = (1 + cos 2) / 2.

Let

sin^4 x = A + B cos 2x + C cos 4x ................... (1)

We have

sin^4 x = ( sin^2 x )^2 = [(1 - cos 2x) / 2]^2
     = (1/4) * [ 1 - 2 cos 2x + cos^2 2x ]
       = (1/4) * { 1 - 2 cos 2x + [ (1 + cos 4x) / 2]}
           = (1/4) * [ ( 2 - 4 cos 2x + 1 + cos 4x ) / 2 ]
   = (1/8) * [ 3 - 4 cos 2x + cos 4x ]
           = (3/8) - (1/2).cos 2x + (1/8).cos 4x ..........................(2)

Comparing (1) with (2),

A = 3/8, B = -1/2 and C = 1/8.