Question

Simplify \( \frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} \).

• A. \( 2 \sec A \)
• B. \( 2 \csc A \)
• C. \( \sec A \)
• D. \( \csc A \)

Hint:

To simplify trigonometric expressions, always look for common factors and use standard trigonometric identities to combine terms.

Answer 1

The Correct Option is B

Step 1: Simplifying the expression.
We start by simplifying the given expression \( \frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} \). To combine the two terms, we find a common denominator: \[ \frac{\sin A}{1 + \cos A} + \frac{1 + \cos A}{\sin A} = \frac{\sin^2 A + (1 + \cos A)^2}{(1 + \cos A)\sin A} \] Step 2: Expanding and simplifying.
The numerator becomes: \[ \sin^2 A + (1 + \cos A)^2 = \sin^2 A + 1 + 2\cos A + \cos^2 A = 2 + 2\cos A \] Thus, the expression becomes: \[ \frac{2(1 + \cos A)}{(1 + \cos A)\sin A} = \frac{2}{\sin A} \] Step 3: Conclusion.
The simplified expression is \( \frac{2}{\sin A} = 2 \csc A \). Therefore, the correct answer is (B) \( 2 \csc A \).