$\displaystyle

\begin{aligned}\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}

{{\theta + \tan \theta }} &= \mathop {\lim }\limits_{\theta \to 0} \left\{ {\frac{{\sin \theta }}

{\theta }\cdot \frac{\theta }

{{\theta +\tan\theta }}} \right\}\\

&= \mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}

{\theta } \cdot \mathop {\lim }\limits_{\theta\to 0} \frac{1}

{{1 + \dfrac{{\tan \theta }}

{\theta }}}\\

&= \frac{1}

{2}.\end{aligned}$

This is equal to THP's work. While I was posting he beat me to it.

(Taylor series solution will make no sense to Coco87.)