First, let's simplify the given trigonometric expression. We know that the cotangent function is the reciprocal of the tangent function, so cot(3π/4) is equivalent to 1/tan(3π/4). The tangent function is equal to sin/cos, so tan(3π/4) = sin(3π/4) / cos(3π/4) = -1.

Therefore, cot(3π/4) = 1/(-1) = -1.

Now, we need to calculate sin(5π/4) and cos(7π/4) to simplify the expression further.

sin(5π/4) = sin(π/4) = sqrt(2)/2
cos(7π/4) = cos(π/4) = sqrt(2)/2

Substitute these values back into the original expression:

-1 (sqrt(2)/2) (sqrt(2)/2) = -1 * (2/4) = -1/2

Therefore, the simplified form of the expression 10 cot(3π/4) sin(5π/4) cos(7π/4) is -5.