Proof:

Let's simplify the given expression step by step to determine if it equals zero.

Step 1: Expand the expression
(sina cosb)×(sina-sinb)-(cos^2b-cos^2a)

Expanding the first term, we get:
(sina^2 cosb - sina sinb cosb) - (cos^2b - cos^2a)

Step 2: Rearrange terms
Now, let's rearrange the terms to group similar ones together:
sina^2 cosb - sina sinb cosb - cos^2b + cos^2a

Step 3: Factor out common terms
Next, we can factor out common terms from each pair of terms:
sina^2 cosb - cos^2b - sina sinb cosb + cos^2a

Step 4: Use trigonometric identities
We can simplify the expression further by using some trigonometric identities. One such identity is sin^2θ + cos^2θ = 1. Rearranging this equation, we get sin^2θ = 1 - cos^2θ.

Using this identity, we can rewrite our expression as:
(1 - cos^2a) cosb - cos^2b - sina sinb cosb + cos^2a

Step 5: Combine like terms
Now, let's combine the like terms:
cosb - cos^2a cosb - cos^2b - sina sinb cosb + cos^2a

Step 6: Factor out common factors
Factor out cosb from the first and third terms, and factor out -1 from the second and fourth terms:
cosb(1 - cos^2a - sina sinb) - (cos^2b - cos^2a)

Step 7: Simplify further
Using the identity sin^2θ = 1 - cos^2θ again, we can rewrite the expression as:
cosb(sin^2a - sina sinb) - (cos^2b - cos^2a)

Step 8: Use a trigonometric identity
Another trigonometric identity that can be useful here is sin(A + B) = sinA cosB + cosA sinB. Rearranging this identity, we get sinA sinB = (sinA cosB - cosA sinB).

Using this identity, we can rewrite our expression as:
cosb(sin^2a - sin(a+b) sinb) - (cos^2b - cos^2a)

Step 9: Simplify the expression further
Expanding the second term, we get:
cosb(sin^2a - sinacosb - sinbcos^2a) - (cos^2b - cos^2a)

Step 10: Combine like terms
Let's combine the like terms:
cosb(sin^2a - sinacosb - sinbcos^2a) - cos^2b + cos^2a

Step 11: Rearrange terms
Rearranging the terms, we get:
cosb(sin^2a - sinbcos^2a - sinacosb) + cos^2a - cos^2b

Step 12: Factor out common terms
Now, let's factor out common terms from each pair of terms:
sin^2a(cosb - cos