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

Simplifying the LHS:
We start by expanding the expression using the trigonometric identities:

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

Expanding the first term using the distributive property:

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

Using the identity sin^2 x = 1 - cos^2 x, we can rewrite the first term:

LHS = (1 - cos^2 a) cos b - sin a cos^2 b - (cos^2 b - cos^2 a)

Now, let's simplify each term separately:

Term 1: (1 - cos^2 a) cos b = cos b - cos^2 a cos b

Term 2: -sin a cos^2 b = -cos^2 b sin a

Term 3: -(cos^2 b - cos^2 a) = cos^2 a - cos^2 b

Substituting these simplified terms back into the original expression:

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

Now, let's combine like terms:

LHS = cos b - cos^2 a cos b - cos^2 b sin a - cos^2 a + cos^2 b

Rearranging the terms:

LHS = cos b - cos^2 a cos b - cos^2 a + cos^2 b - cos^2 b sin a

Simplifying further:

LHS = cos b - cos^2 a + cos^2 b - cos^2 a cos b - cos^2 b sin a

Now, let's factor out common terms:

LHS = (1 - cos^2 a) - cos^2 a cos b + (1 - cos^2 b) - cos^2 b sin a

Using the identity sin^2 x = 1 - cos^2 x again:

LHS = sin^2 a - cos^2 a cos b + sin^2 b - cos^2 b sin a

Simplifying:

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

Applying the identity sin^2 x + cos^2 x = 1:

LHS = 1 - cos a cos b (cos a + cos b)

Proving the Equation:
To prove that LHS = 0, we need to show that 1 - cos a cos b (cos a + cos b) = 0.

Let's consider two cases:

Case 1: If cos a = 0 or cos b = 0, then the equation is satisfied since any term multiplied by zero results in zero.

Case 2: If cos a and cos b are not zero, then we can divide both sides of the equation by cos a cos b (cos a + cos b):

1 - cos a cos b (cos a + cos b) = 0

Dividing by cos a cos

To prove the equation (sina cosb)×(sina-cosb)-(cos^2b-cos^2a) = 0, we can start by expanding the expression and simplifying it step by step.

Expanding the expression:
(sina cosb)×(sina-cosb)-(cos^2b-cos^2a)

Distributing sina cosb:
(sina^2 cosb - sina cos^2b) - (cos^2b - cos^2a)

Simplifying the expression:
sina^2 cosb - sina cos^2b - cos^2b + cos^2a

Grouping similar terms:
sina^2 cosb - sina cos^2b - (cos^2b - cos^2a)

Using the identity sin^2x + cos^2x = 1:
sina^2 cosb - sina cos^2b - (1 - cos^2a)

Expanding the brackets:
sina^2 cosb - sina cos^2b - 1 + cos^2a

Combining like terms:
sina^2 cosb - sina cos^2b + cos^2a - 1

Now, to prove that this expression is equal to 0, we need to show that it simplifies further to 0.

Using the identity sin^2x + cos^2x = 1:
(1 - cos^2b) cosb - sina cos^2b + cos^2a - 1

Expanding the brackets:
cosb - cos^3b - sina cos^2b + cos^2a - 1

Combining like terms:
cosb - cos^3b - sina cos^2b + cos^2a - 1

Rearranging the terms:
cosb - sina cos^2b + cos^2a - cos^3b - 1

Using the identity sin^2x + cos^2x = 1:
cosb - (1 - sin^2b) cos^2b + cos^2a - cos^3b - 1

Expanding the brackets:
cosb - cos^2b + sin^2b cos^2b + cos^2a - cos^3b - 1

Combining like terms:
cosb - cos^2b + sin^2b cos^2b + cos^2a - cos^3b - 1

Rearranging the terms:
cosb - cos^2b + sin^2b cos^2b - cos^3b + cos^2a - 1

Using the identity sin^2x + cos^2x = 1:
cosb - cos^2b + (1 - cos^2b) cos^2b - cos^3b + cos^2a - 1

Expanding the brackets:
cosb - cos^2b + cos^2b - cos^4b - cos^3b + cos^