TITLE: Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
QUESTION [14 upvotes]: Dealing with a physics Problem I get the following equation to solve for $\alpha$
$1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$
Putting this in Mathematica gives the result:
$a==2 \text{ArcTan}\left[\frac{1-m}{1+m}\right]$
However I am unable to get this result myself. No matter what I try normal equation transformations or rewriting the equation with the complex e-Function ..., everything fails. Even going the other Direction from Mathematica's Solution to my original equation resulted in nothing sensible. 
Any help of how to do do this transformation is very much appreciated.
Thanks in advance

REPLY [5 votes]: Another:
$$
\begin{eqnarray}
\frac{1-\sin\alpha}{\cos\alpha} &=& m \\
\frac{1-\cos\left(\frac{\pi}{2}-\alpha\right)}{\sin\left(\frac{\pi}{2}-\alpha\right)} &=&m \\
\tan\frac{\frac{\pi}{2}-\alpha}{2} = \tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right) &=& m \\
\frac{\tan\frac{\pi}{4}-\tan\frac{\alpha}{2}}{1+\tan\frac{\pi}{4} \tan\frac{\alpha}{2}} = \frac{1-\tan\frac{\alpha}{2}}{1+\tan\frac{\alpha}{2}} &=& m \\
\Longrightarrow \tan\frac{\alpha}{2} &=& \frac{1-m}{1+m}
\end{eqnarray}
$$