Question

The expression $ \frac{2 \tan A}{1 - \cot A} + \frac{2 \cot A}{1 - \tan A} $ \text{ can be written as}

• A. \( \sin 2A + \cos 2A \)
• B. \( 2 \sec A \csc A + 2 \)
• C. \( \tan 2A + \cot 2A \)
• D. \( \sec 2A + \csc 2A \)

Hint:

To simplify complex trigonometric expressions, convert cotangent and tangent into their sine and cosine equivalents. Use known trigonometric identities to further simplify.

Answer 1

The Correct Option is B

The given expression is: \[ \frac{2 \tan A}{1 - \cot A} + \frac{2 \cot A}{1 - \tan A} \] We know that: \[ \cot A = \frac{1}{\tan A} \] Now, simplify the two terms separately: \[ \frac{2 \tan A}{1 - \cot A} = \frac{2 \tan A}{1 - \frac{1}{\tan A}} = \frac{2 \tan A}{\frac{\tan A - 1}{\tan A}} = 2 \sec A \csc A \] Similarly, for the second term: \[ \frac{2 \cot A}{1 - \tan A} = 2 \sec A \csc A \] Now, adding both terms: \[ 2 \sec A \csc A + 2 \] Thus, the simplified expression is: \[ 2 \sec A \csc A + 2 \]