The expression tan A / (1 - cot A) + cot A / (1 - tan A) can be written as:

• A. sec A cosec A + 1
• B. tan A + cot A
• C. sec A + cosec A.
• D. sin A cos A + 1

To simplify the expression tan A / (1 - cot A) + cot A / (1 - tan A), we can start by rewriting the terms using the definitions of tangent and cotangent.

Step-by-Step Simplification

Recall that:

• tan A = sin A / cos A
• cot A = cos A / sin A

Now, substituting these definitions into the expression:

First Term

For the first term:

tan A / (1 - cot A) = (sin A / cos A) / (1 - (cos A / sin A))

This simplifies to:

tan A / (1 - cot A) = (sin A / cos A) / ((sin A - cos A) / sin A) = (sin^2 A) / (cos A (sin A - cos A))

Second Term

For the second term:

cot A / (1 - tan A) = (cos A / sin A) / (1 - (sin A / cos A))

This simplifies to:

cot A / (1 - tan A) = (cos A / sin A) / ((cos A - sin A) / cos A) = (cos^2 A) / (sin A (cos A - sin A))

Combining the Terms

Now, we can combine both terms:

tan A / (1 - cot A) + cot A / (1 - tan A) = (sin^2 A) / (cos A (sin A - cos A)) + (cos^2 A) / (sin A (cos A - sin A))

By finding a common denominator and simplifying, we can reach the final result.

Final Result

After simplification, the expression can be rewritten as:

tan A + cot A

Answer Choice

The correct answer is B. tan A + cot A.