1/2 + 3sin²x - 3sinx = 1/2cos2x

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To solve this equation, we need to simplify the left side first.

1/2 + 3sin²x - 3sinx

= 1/2 + 3sinx(sinx - 1)

= 1/2 + 3sinx(-cosx)

= 1/2 - 3sinx*cosx

Now, we can rewrite the right side of the equation as:

1/2cos2x = 1/2(2cos²x - 1) = cos²x - 1/2

So, the equation becomes:

1/2 - 3sinx*cosx = cos²x - 1/2

Rearranging the terms, we get:

cos²x - 3sinx*cosx - 1 = 0

Now, we can apply the double angle identity for cosine:

cos²x - 3sinx*cosx - 1 = 0

cos²x - 3sinx*cosx - 1 = cos(2x) - 1

cos(2x) - 1 = 0

cos(2x) = 1

Now, for cos(2x) = 1, the possible values of x are:

2x = 0

x = 0

Therefore, the solution to the equation is x = 0.