Question about a trigonometry proof? I just want to ask how can you prove that 2α is twice the value of α in the following figure that depicts a proof of an arctangent identity (and likewise, for β as well).


let $$a = \tan^{-1}\left(\frac{1-y}x\right), b = \tan^{-1}\left(\frac{1-x}y\right) .$$  then 
$$\begin{align}\tan(a+b) &= \frac{\tan a + \tan b}{1- \tan a \tan b} \\
&= \frac{\frac{1-y}x+\frac{1-x}y}{1-\frac{1-y}x \frac{1-x}y} \\
&= \frac{y+x-y^2-x^2}{xy-(1-x-y+xy)} \\
&=\frac{(x+y)(1-x-y)}{-(1-x-y)} \\
&= -x-y\end{align}$$
don't you need $x+y = -1$  for $a+b = \pi/4?$
