Given: cos A - sin A = 1

To prove: cos A sin A = 1 or -1

Proof:

We are given the equation cos A - sin A = 1.

Step 1: Rearrange the equation
We can rearrange the equation as cos A = 1 + sin A.

Step 2: Square both sides
To simplify the equation, we can square both sides of the equation.

(cos A)^2 = (1 + sin A)^2

Step 3: Expand and simplify
Expanding the equation, we get:

cos^2 A = 1 + 2sin A + (sin A)^2

Simplifying further, we have:

cos^2 A = 1 + 2sin A + sin^2 A

Step 4: Use the identity sin^2 A + cos^2 A = 1
We know that sin^2 A + cos^2 A = 1. We can substitute this identity into the above equation.

1 - sin^2 A = 1 + 2sin A + sin^2 A

Simplifying further, we have:

1 - 1 = 2sin A + 2sin^2 A

0 = 2sin A (1 + sin A)

Step 5: Solve for sin A
Since sin A cannot be zero (as it is in the denominator of the equation), we can divide both sides of the equation by 2(1 + sin A).

0 = sin A

Conclusion:
From the above equation, we can see that sin A = 0. Therefore, cos A sin A = 1 x 0 = 0.

So, cos A sin A is not equal to 1 or -1.