To solve the equation sin(3x)sin(x) - cos(7x)cos(5x) = 0, we can use the product-to-sum identities for sine and cosine, which state that:

sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]
cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)]

Applying these identities to the given equation, we have:

1/2[cos(3x-x) - cos(3x+x)] - 1/2[cos(7x+5x) + cos(7x-5x)] = 0
1/2[cos(2x) - cos(4x)] - 1/2[cos(12x) + cos(2x)] = 0
1/2cos(2x) - 1/2cos(4x) - 1/2cos(12x) - 1/2cos(2x) = 0
-1/2cos(4x) - 1/2cos(12x) = 0

Now we can simplify this to:

-1/2cos(4x) - 1/2cos(12x) = 0
cos(4x) + cos(12x) = 0

However, this equation does not immediately yield a solution since the sum of two cosine functions may not necessarily be zero. You may need to further simplify or revise your approach to finding solutions.