To prove this equation, we need to use the sum-to-product trigonometric identity, which states that:

cos(a)cos(b) = 0.5[cos(a+b) + cos(a-b)]

Therefore, we can rewrite the left side of the equation as:

cos(x)cos(3x) = 0.5[cos(4x) + cos(2x)]

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

cos(5x)cos(7x) = 0.5[cos(12x) + cos(-2x)]

Since cos(-θ) = cos(θ), we have:

cos(-2x) = cos(2x)

Therefore, the right side can be simplified as:

cos(5x)cos(7x) = 0.5[cos(12x) + cos(2x)]

Now, we can see that both sides of the equation are equal to 0.5[cos(12x) + cos(2x)], which means:

cos(x)cos(3x) = cos(5x)cos(7x)

Therefore, the equation is proven to be true.