Question

If \( \sqrt{\frac{1 + \cos A}{1 - \cos A}} \) is, then the correct option is:

• A. \( \csc A - \cot A \)
• B. \( \csc A + \cot A \)
• C. \( \csc A \cdot \cot A \)
• D. \( \sin A \cdot \tan A \)

Hint:

To simplify trigonometric expressions involving square roots, consider using half angle formulas for \( \cos A \) and \( \sin A \).

Answer 1

The Correct Option is A

The given expression is \( \sqrt{\frac{1 + \cos A}{1 - \cos A}} \). By applying the standard identity \( 1 + \cos A = 2 \cos^2 \frac{A}{2} \) and \( 1 - \cos A = 2 \sin^2 \frac{A}{2} \), we can simplify as: \[ \sqrt{\frac{2 \cos^2 \frac{A}{2}}{2 \sin^2 \frac{A}{2}}} = \frac{\cos \frac{A}{2}}{\sin \frac{A}{2}} = \csc A - \cot A \] Thus, the correct answer is \( \csc A - \cot A \).