If A is an acute angle, which of the following is equal to \{sin A}/{1 + cos A}?

Question

Accepted Answer

Given expression: $\frac{\sin A}{1 + \cos A}$

Multiply numerator and denominator by $1 - \cos A$:
$$
\frac{\sin A}{1 + \cos A} \times \frac{1 - \cos A}{1 - \cos A} = \frac{\sin A (1 - \cos A)}{(1 + \cos A)(1 - \cos A)}
$$

Using the identity $(1 + \cos A)(1 - \cos A) = 1 - \cos^2 A = \sin^2 A$, we get:
$$
\frac{\sin A (1 - \cos A)}{\sin^2 A} = \frac{1 - \cos A}{\sin A}
$$

Hence, $\frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$.

Answer

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