3sin^2(3x)-2,5sin(6x)+1=0

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Let's first simplify the given equation by using the trigonometric identity sin^2(x) = 1/2 - 1/2cos(2x):

3(1/2 - 1/2cos(6x)) - 5sin(6x) + 1 = 0
3/2 - 3/2cos(6x) - 5sin(6x) + 1 = 0
5/2 - 3/2cos(6x) - 5sin(6x) = 0
3/2cos(6x) + 5sin(6x) = 5/2

Now, we can use the cosine sum formula to rewrite the left side of the equation:

3/2(cos(6x)cos(0) - sin(6x)sin(0)) + 5sin(6x) = 5/2
3/2(cos(6x)1 - sin(6x)0) + 5sin(6x) = 5/2
3/2cos(6x) + 5sin(6x) = 5/2

Therefore, the given equation simplifies to 3/2cos(6x) + 5sin(6x) = 5/2.