$\dfrac{\sin \theta}{(1+ \cos \theta)}$ is equal to

Multiply both numerator and denominator by $(1 - \cos \theta)$:

$\dfrac{\sin \theta}{1+ \cos \theta} \times \dfrac{1 - \cos \theta}{1 - \cos \theta}$

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The denominator becomes:

$(1 + \cos \theta)(1 - \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta$

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The numerator becomes:

$\sin \theta (1 - \cos \theta) = \sin \theta - \sin \theta \cos \theta$

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Therefore:

$\dfrac{\sin \theta - \sin \theta \cos \theta}{\sin^2 \theta}$

$= \dfrac{\sin \theta(1 - \cos \theta)}{\sin^2 \theta}$

$= \dfrac{1 - \cos \theta}{\sin \theta}$