To simplify the given expression, we will use the trigonometric identity for the cosine of the difference of two angles, which states that:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

In this case, a = 5x and b = 3x:

cos(5x - 3x) = cos(5x)cos(3x) + sin(5x)sin(3x)

cos(2x) = cos(5x)cos(3x) + sin(5x)sin(3x)

Since cos(2x) = cos(8x), the simplified expression we were asked to prove is:

cos 5x cos 3x - sin 5x sin 3x = cos 8x