Trigonometric equality: $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$ Can you guys give me a hint on how to proceed with proving this trigonometric equality? I have a feeling I need to use the half angle identity for $\tan \frac{\theta}{2}$. The stuff I have tried so far(multiplying numerator and denominator by $1 + \sin A - \cos A$) has lead to a dead end.

Prove that,
  $$
\dfrac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \dfrac{A}{2}
$$


$\textbf{Method 1.}$ You have 
\begin{align*}
\frac{1+ \sin{A} - \cos{A}}{1+\sin{A} + \cos{A}} &= \frac{ \cos^{2}\frac{A}{2} + \sin^{2}\frac{A}{2} + 2\cdot \sin\frac{A}{2}\cdot\cos\frac{A}{2} - \Bigl(\cos^{2}\frac{A}{2} - \sin^{2}\frac{A}{2}\Bigr)}{\cos^{2}\frac{A}{2} + \sin^{2}\frac{A}{2} + 2\cdot \sin\frac{A}{2}\cdot\cos\frac{A}{2} + \Bigl(\cos^{2}\frac{A}{2} -\sin^{2}\frac{A}{2}\Bigr)} \\ &=\frac{ \Bigl(\cos\frac{A}{2} + \sin\frac{A}{2}\Bigr)^{2} - \Bigl(\cos\frac{A}{2}+\sin\frac{A}{2}\Bigr) \cdot \Bigl(\cos\frac{A}{2} - \sin\frac{A}{2}\Bigr)}{\Bigl(\cos\frac{A}{2} + \sin\frac{A}{2}\Bigr)^{2} + \Bigl(\cos\frac{A}{2}+\sin\frac{A}{2}\Bigr) \cdot \Bigl(\cos\frac{A}{2} - \sin\frac{A}{2}\Bigr)} \\ &= \frac{ \cos\frac{A}{2} + \sin\frac{A}{2} - \cos\frac{A}{2} + \sin\frac{A}{2}}{\cos\frac{A}{2} + \sin\frac{A}{2} + \cos\frac{A}{2} -\sin\frac{A}{2}} \qquad \Bigl[ \text{cancelling}\ \Bigl(\small\cos\frac{A}{2} +\sin\frac{A}{2}\Bigr) \ \Bigr] \\ &= \tan\frac{A}{2}
\end{align*}
$\textbf{Method 2.}$ Here is another way of seeing this. By using something called Componendo and Dividendo.


*

*http://en.wikipedia.org/wiki/Componendo_and_dividendo
Let $$\frac{1+\sin{A} -\cos{A}}{1+\sin{A} + \cos{A}} = \frac{k}{1}$$ Now applying componendo and dividendo we get $$\frac{1+ \sin{A}}{-\cos{A}} = \frac{k+1}{k-1} \Longrightarrow \frac{\cos\frac{A}{2}+\sin\frac{A}{2}}{-\cos\frac{A}{2}+\sin\frac{A}{2}} = \frac{k+1}{k-1} \qquad (1)$$ 
Again using componendo dividendo on $(1)$ we get $$\frac{\cos\frac{A}{2}+\sin\frac{A}{2} -\cos\frac{A}{2}+\sin\frac{A}{2}}{\cos\frac{A}{2}+\sin\frac{A}{2} +\cos\frac{A}{2}-\sin\frac{A}{2}} = \frac{k+1 +k-1}{k+1-k+1} \Longrightarrow k=\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}}=\tan\frac{A}{2}$$
$\textbf{Method 3.}$ Another way of looking into this would be,
\begin{align*}
\frac{1+\sin{A}-\cos{A}}{1+\sin{A}+\cos{A}} =\frac{(1-\cos{A})+\sin{A}}{(1+\cos{A}) + \sin{A}} &= \frac{ 2\:\sin^{2}\frac{A}{2} + 2\cdot\sin\frac{A}{2}\cdot\cos\frac{A}{2}}{2\:\cos^{2}\frac{A}{2} + 2 \cdot\sin\frac{A}{2}\cdot \cos\frac{A}{2}} \\ &= \frac{2\:\sin\frac{A}{2}}{2\:\cos\frac{A}{2}} \cdot \Biggl(\frac{\cos\frac{A}{2}+\sin\frac{A}{2}}{\cos\frac{A}{2} + \sin\frac{A}{2}}\Biggr) \\ &= \tan\frac{A}{2}
\end{align*}
$\textbf{Method 4.}$ My younger brother mentioned about this method. Multiplying numerator and deominator by $(1+\sin{A}-\cos{A})$ we get 
\begin{align*}
\frac{1+\sin{A}-\cos{A}}{1+\sin{A}+\cos{A}} \times \frac{1+\sin{A}-\cos{A}}{1+\sin{A}-\cos{A}} &= \frac{(1+\sin{A}-\cos{A})^{2}}{(1+\sin{A})^{2}-\cos^{2}{A}} \\ &= \frac{2 + 2\: \sin{A} -2\cdot\sin{A}\cdot \cos{A} -2\cos{A}}{2\: \sin{A} + 2\sin^{2}{A}} \\ &= \frac{(2-2\cos{A}) \cdot (1+\sin{A})}{2\:\sin{A} \cdot (1+\sin{A})} \\  &=\frac{1-\cos{A}}{\sin{A}} = \frac{2\: \sin^{2}\frac{A}{2}}{2\cdot \sin\frac{A}{2} \cdot \cos\frac{A}{2}} \\ &=\tan\frac{A}{2}
\end{align*}
Oh once again when I looked at the question I realize that you attempted this method. Hopefully now it's clear.
