If \( A \) is an acute angle, which of the following is equal to: \[ \frac{\sin A}{1 + \cos A} \, ? \]
\( \frac{1 - \cos A}{\sin A} \)
We need to simplify \( \frac{\sin A}{1 + \cos A} \) and check which option it matches.
Step 1: Multiply by the conjugate
To simplify \( \frac{\sin A}{1 + \cos A} \), multiply the numerator and denominator by the conjugate of \( 1 + \cos A \), which is \( 1 - \cos A \):
\[ \frac{\sin A}{1 + \cos A} \times \frac{1 - \cos A}{1 - \cos A} = \frac{\sin A (1 - \cos A)}{(1 + \cos A)(1 - \cos A)}. \]
Step 2: Simplify the denominator
Using the difference of squares formula:
\[ (1 + \cos A)(1 - \cos A) = 1 - \cos^2 A. \]
Since \( \sin^2 A = 1 - \cos^2 A \), the denominator becomes:
\[ \sin^2 A. \]
Thus, the expression is now:
\[ \frac{\sin A (1 - \cos A)}{\sin^2 A}. \]
Step 3: Simplify the numerator
Split the terms:
\[ \frac{\sin A (1 - \cos A)}{\sin^2 A} = \frac{\sin A}{\sin^2 A} \cdot (1 - \cos A). \]
Simplify \( \frac{\sin A}{\sin^2 A} \) as \( \frac{1}{\sin A} \):
\[ \frac{\sin A (1 - \cos A)}{\sin^2 A} = \frac{1 - \cos A}{\sin A}. \]
Step 4: Compare with options
From the given options, the simplified expression matches:
Option 1: \( \frac{1 - \cos A}{\sin A} \)
Final Answer
The correct option is \( \boxed{3} \).
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