**Proof:**

To prove that Sec(A) tan(A) - 1 / tan(A) - Sec(A) = Cos(A) / 1 - Sin(A), we will start by simplifying the left side of the equation and then simplify the right side of the equation. We will then show that both sides are equal.

**Simplifying the left side:**

Step 1:
We will simplify the left side of the equation by finding the common denominator for the terms Sec(A) and tan(A).

Sec(A) tan(A) - 1 / tan(A) - Sec(A)

Step 2:
To find the common denominator, we multiply the first term (Sec(A) tan(A) - 1) by tan(A) and the second term (tan(A) - Sec(A)) by Sec(A).

(tan(A) * Sec(A) * tan(A) - 1 * tan(A)) / (tan(A) * tan(A) - Sec(A) * tan(A))

Step 3:
Simplify further by applying the trigonometric identity tan(A) = Sin(A) / Cos(A) and Sec(A) = 1 / Cos(A).

(Sin(A) / Cos(A) * Sec(A) * Sin(A) / Cos(A) - tan(A) / Cos(A)) / (tan(A) * Sin(A) / Cos(A) - Sec(A) * Sin(A) / Cos(A))

Step 4:
Multiply the numerators and denominators and simplify the expression.

(Sin^2(A) / Cos^2(A) - tan(A) / Cos(A)) / (tan(A) * Sin(A) / Cos^2(A) - Sin(A) / Cos(A))

Step 5:
Combine the fractions by finding a common denominator for the terms in the numerator and the terms in the denominator.

(Sin^2(A) - tan(A) * Cos(A)) / (tan(A) * Sin(A) - Sin(A) * Cos(A))

Step 6:
Factor out Sin(A) from the numerator and denominator.

Sin(A) * (Sin(A) - tan(A) * Cos(A)) / Sin(A) * (tan(A) - Cos(A))

Step 7:
Cancel out the common term Sin(A) from the numerator and denominator.

(Sin(A) - tan(A) * Cos(A)) / (tan(A) - Cos(A))

**Simplifying the right side:**

Step 8:
We will simplify the right side of the equation.

Cos(A) / (1 - Sin(A))

Step 9:
Multiply the numerator and denominator by the conjugate of (1 - Sin(A)) to simplify further.

Cos(A) * (1 + Sin(A)) / (1 - Sin^2(A))

Step 10:
Apply the trigonometric identity Sin^2(A) + Cos^2(A) = 1.

Cos(A) * (1 + Sin(A)) / Cos^2(A)

Step 11:
Cancel out the common term Cos(A) from the numerator and denominator.

(1 + Sin(A)) / Cos(A)

**Comparing both sides:**

Step 12:
Comparing the simplified left side (from step 7) and the simplified right side (from step 11), we observe that they are equal.

(Sin(A) - tan(A) * Cos(A)) / (tan(A) - Cos(A)) = (