I have been scratching my head all day over this question for one of my assignments. I haven't made any progress and I'm at the point of giving up. Here's what I need help with.
Start by gluing together a sequence of right angled triangles as shown in the
following diagram.
The angles $\theta_i$, $i = 1, 2, 3, \dots$ can be freely chosen. This construction produces a sequence of points $(x,y)_n$ in the plane. Show that \begin{align} x_{n+1} &= x_n - y_n \tan \theta_{n+1} \tag{1} \\ y_{n+1} &= y_n + x_n \tan \theta_{n+1} \tag{2} \end{align}
I have no idea how to even begin to work this out. Any help will be greatly appreciated!
To your picture, I have added some extras as hint. Hope that helps.