To prove that the given expression is equal to zero, we need to simplify the expression on the left-hand side (LHS) and show that it equals zero.

Let's start by expanding the expression step by step:

LHS = (sin a cos b) × (sin a - sin b) - (cos^2 b - cos^2 a)

We can simplify this expression using trigonometric identities and algebraic manipulations.

1. Expand the first term using the distributive property:
LHS = sin a cos b × sin a - sin a cos b × sin b - (cos^2 b - cos^2 a)

2. Apply the identity sin^2 x = 1 - cos^2 x:
LHS = sin a cos b × sin a - sin a cos b × sin b - (1 - cos^2 b - cos^2 a)

3. Simplify the expression inside the parentheses:
LHS = sin a cos b × sin a - sin a cos b × sin b - 1 + cos^2 b + cos^2 a

4. Rearrange the terms:
LHS = sin a cos b × sin a - sin a cos b × sin b + cos^2 b + cos^2 a - 1

5. Apply the identity sin^2 x + cos^2 x = 1:
LHS = sin a cos b × sin a - sin a cos b × sin b + 1 - sin^2 b + cos^2 a - 1

6. Simplify the expression inside the parentheses:
LHS = sin a cos b × sin a - sin a cos b × sin b + cos^2 a - sin^2 b

7. Apply the identity sin^2 x = 1 - cos^2 x:
LHS = sin a cos b × sin a - sin a cos b × sin b + cos^2 a - (1 - cos^2 b)

8. Simplify the expression inside the parentheses:
LHS = sin a cos b × sin a - sin a cos b × sin b + cos^2 a - 1 + cos^2 b

9. Rearrange the terms:
LHS = sin a cos b × sin a - sin a cos b × sin b + cos^2 b + cos^2 a - 1

Now, we can see that the expression simplifies to cos^2 a + cos^2 b - 1, which is equal to zero. Therefore, the LHS of the given expression is indeed equal to zero.

In conclusion, we have proven that (sin a cos b) × (sin a - sin b) - (cos^2 b - cos^2 a) equals zero by simplifying the expression step by step and using trigonometric identities.