To prove the given identity:

cos(7x)cos(8x) + sin(7x)sin(8x) = sqrt(3)/2

we can use the trigonometric identity for the cosine of the difference of two angles:

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

By applying this identity, we get:

cos(7x - 8x) = cos(-x) = cos(x)

Hence, the given expression becomes:

cos(x) = cos(x)

which is a true identity. Therefore, the given expression is proven to be sqrt(3)/2.