To solve this trigonometric equation:

sin(15x) sin(3x) + cos(7x) cos(11x) = 0

We can use the angle difference identity for cosine:

cos(a) * cos(b) = (1/2)[cos(a-b) + cos(a+b)]

Applying this identity to the given equation, we get:

[sin(12x) - sin(18x)]/2 + [cos(4x) + cos(18x)]/2 = 0

solving for the equation, we have:
sin(12x) - sin(18x) + cos(4x) + cos(18x) = 0

Rearranging terms, we get:

sin(12x) + cos(4x) = sin(18x) - cos(18x)

Using the sum to product identities:

sin(12x) + cos(4x) = 2sin[(12x + 4x)/2] * cos[(12x - 4x)/2]

sin(18x) - cos(18x) = 2sin[18x/2] * cos[-18x/2]

Simplifying:

[2sin(8x)cos(4x)] = [2sin(9x)cos(9x)]

sin(8x)cos(4x) = sin(9x)cos(9x)

sin(8x + 4x) = sin(9x)

sin(12x) = sin(9x)

Since sine functions are equal, the angles themselves must also be equal:

12x = 9x

Solving for x, we get:

12x - 9x = 0

3x = 0

x = 0

Therefore, the solution to the given trigonometric equation sin(15x) sin(3x) + cos(7x) cos(11x) = 0 is x = 0.