(1-cos6x)*cos2x=sin^2 3x

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To solve this trigonometric equation, we can use the double angle formula for cosine:

cos(2x) = 2cos^2(x) - 1

Substitute this formula into the equation:

(1 - cos(6x)) * (2cos^2(x) - 1) = sin^2(3x)

Expand the left side:

2cos^2(x) - cos(6x) - 2cos^2(x) + 1 = sin^2(3x)

Simplify:

Since sin^2(x) = 1 - cos^2(x), we can rewrite the equation as:

cos(6x) = cos^2(3x)

Apply the double angle formula for cosine again to the right side:

cos(6x) = (1 + cos(6x))/2

Rearranging the equation gives:

2cos(6x) = 1 + cos(6x)

cos(6x) = 1

As a result, this equation simplifies to:

cos(6x) = 1

Since the range of cosine is -1 to 1, the only solution to this equation is cos(6x) = 1, which implies that 6x must be a multiple of 2π:

6x = 2πn, where n is any integer

x = πn/3, where n is any integer

Therefore, the general solution to the equation is x = πn/3, where n is an integer.