Question

The expression \( \frac{\tan A + \cot A}{1 - \cot A} \) can be written as:

• A. \( \sin A \cos A + 1 \)
• B. \( \sec A \csc A + 1 \)
• C. \( \tan A + \cot A \)
• D. \( \sec A + \csc A \)

Hint:

When simplifying trigonometric expressions, try converting all terms into sine and cosine to make the simplification easier.

Answer 1

The Correct Option is B

We are asked to simplify the expression: \[ \frac{\tan A + \cot A}{1 - \cot A} \] Step 1: Rewrite in terms of sine and cosine We know the following identities: \[ \tan A = \frac{\sin A}{\cos A}, \quad \cot A = \frac{\cos A}{\sin A} \] Substitute these into the original expression: \[ \frac{\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}}{1 - \frac{\cos A}{\sin A}} \] Step 2: Simplify the expression To simplify the numerator: \[ \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} = \frac{1}{\sin A \cos A} \] For the denominator: \[ 1 - \frac{\cos A}{\sin A} = \frac{\sin A - \cos A}{\sin A} \] Now, the expression becomes: \[ \frac{\frac{1}{\sin A \cos A}}{\frac{\sin A - \cos A}{\sin A}} = \frac{1}{\sin A \cos A} \times \frac{\sin A}{\sin A - \cos A} \] Step 3: Recognize the result Simplifying the expression leads us to the form \( \sec A \csc A + 1 \), which is the correct answer.