Solution:

To simplify the expression sin(theta)cos^3(theta) - cos(theta)sin^3(theta), we can use the trigonometric identity for the product of two sines:

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

Using the identity:
Let A = theta and B = 3theta, we have:

sin(theta)sin(3theta) = (1/2)(cos(theta - 3theta) - cos(theta + 3theta))
= (1/2)(cos(-2theta) - cos(4theta))

Expanding the terms:
cos(-2theta) = cos(2theta) (since cos(-x) = cos(x))
cos(4theta) = cos(2(2theta)) (since cos(2x) = cos^2(x) - sin^2(x))

Substituting the values back into the expression:
sin(theta)sin(3theta) = (1/2)(cos(2theta) - cos(2(2theta)))
= (1/2)(cos(2theta) - (cos^2(2theta) - sin^2(2theta)))

Using the identity:
cos^2(x) - sin^2(x) = cos(2x)

Substituting the value back into the expression:
sin(theta)sin(3theta) = (1/2)(cos(2theta) - cos(2theta))
= 0

Therefore, sin(theta)cos^3(theta) - cos(theta)sin^3(theta) simplifies to 0.

=sin theta cos theta (cos² theta - sin² theta)
=(1/2) sin 2 theta cos 2 theta
=(1/4) sin 4 theta