Verify trigonometry equation $\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A$ Sorry for asking so many of these type of questions. 
How would I verify the following trigonometry identity:
$$\frac{\sin A+\tan A}{\cot A+\csc A}=\sin A \tan A.$$
My work is
$$\frac{\sin A + \frac{\sin A}{\cos A}}{\frac{\cos A}{\sin A} + \frac{1}{\sin A}}.$$
Do I have to use a common denominator between the sin and tan to solve the identity?

$\begin{align}
\dfrac{\sin A + \tan A}{\cot A + \csc A} &= \dfrac{\sin A + \dfrac{\sin A}{\cos A}}{\dfrac{\cos A}{\sin A} + \dfrac{1}{\sin A}}\\
&= \dfrac{\dfrac{\sin A}{\cos A}\color{green}{\left(\cos A + 1\right)}}{\dfrac{1}{\sin A}\color{green}{(\cos A + 1)}}\\
&=\dfrac{\sin A}{\cos A} \div \dfrac{1}{\sin A}\\
&= \sin A \tan A
\end{align}$
