Proof:

To prove the given identity: tan(theta) sec(theta) - 1 / tan(theta) - sec(theta) = sin(theta) / cos(theta)

Let's simplify the left side of the equation step by step:

Step 1: Start with the left side of the equation
tan(theta) sec(theta) - 1 / tan(theta) - sec(theta)

Step 2: Take the common denominator for the fraction
(tan(theta) sec(theta) - 1) / (tan(theta) - sec(theta))

Step 3: Factor out -1 from the numerator
(-1)(1 - tan(theta) sec(theta)) / (tan(theta) - sec(theta))

Step 4: Use the identity tan(theta) = sin(theta) / cos(theta) and sec(theta) = 1 / cos(theta)

(-1)(1 - (sin(theta) / cos(theta))(1 / cos(theta))) / (tan(theta) - (1 / cos(theta)))

Step 5: Simplify the expression in the numerator
(-1)(cos(theta) - sin(theta)) / (tan(theta) - (1 / cos(theta)))

Step 6: Simplify the expression in the denominator
tan(theta) - (1 / cos(theta)) = (sin(theta) / cos(theta)) - (1 / cos(theta)) = (sin(theta) - 1) / cos(theta)

Step 7: Substitute the simplified denominator into the expression
(-1)(cos(theta) - sin(theta)) / ((sin(theta) - 1) / cos(theta))

Step 8: Multiply the numerator by the reciprocal of the denominator
(-1)(cos(theta) - sin(theta)) * (cos(theta) / (sin(theta) - 1))

Step 9: Distribute and simplify
(-1)(cos^2(theta) - sin(theta)cos(theta)) / (sin(theta) - 1)

Step 10: Use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1
(-1)(1 - sin(theta)cos(theta)) / (sin(theta) - 1)

Step 11: Cancel out the common factor of (sin(theta) - 1)
(-1)(1 - sin(theta)cos(theta)) / (sin(theta) - 1) = -1

Therefore, the left side of the equation simplifies to -1, which is equal to the right side of the equation sin(theta) / cos(theta). Hence, the given identity is proven.