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4 cos² x sin x - 2 sin² x = 3 sin x

*if sinx is 0 then ** x = 2 n π , n = natural number*

*if sinx is not 0 then, cancel sin x on both sides,*

*4 cos² x - 2 sin x = 3*

*4 - 4 sin² x - 2 sin x - 3 =0 *

*4 sin² x + 2 sin x -1 = 0*

*Δ = 4 + 16 = 20*

*sin x = (-2 +- 2√5) / 8 = (-1 +- √5) / 4 = *

__ x = 2 n π + 18 deg or 2 n π (- 54 deg = 126 deg)__

4(1-sin²x)sinx-2sin²x = 3sinx

4sinx-4sin³x-2sin²x = 3sinx

4sin³x+2sin²x-sinx = 0

sinx(4sin²x+2sinx-1) = 0

sinx = 0 or 4sin²x+2sinx-1 = 0

x = 0 or sinx = {-2+-√(2²-4*4*(-1))}/2*4

x = 0 or sinx = (-1+-√5)/4

x = 0 or sinx = (-1-√5)/4 or sinx = (-1+√5)/4

x = 0 or x = sin^-1{(-1-√5)/4} or x = sin^-1{(-1+√5)/4}

there is no boundary condition on x so general solution of x is

x = (nπ+(-1)^n 0) or x = [nπ+(-1)^n sin^-1{(-1-√5)/4}] or (where n is an integer)

x = [nπ+(-1)^n sin^-1{(-1+√5)/4}]

hence

x = nπ or

x = [nπ+(-1)^n sin^-1{(-1-√5)/4}] or

x = [nπ+(-1)^n sin^-1{(-1+√5)/4}]