For a geometric rectangle with arbitrary sides of length $a$, $b$, and $c$.
The following are known:
- $\theta$ where $\frac ac = \tan\theta$
- $\phi$ where $\frac bc = \tan\phi$
Given only $theta$ and $phi$ find $x$ where $$\frac {\sqrt{a^2 + b^2}}c = tan x$$
This can clearly be done with the equation: $$\arctan \sqrt{\tan^2\theta + \tan^2\phi}$$
I don't know a trigonometric identity to simplify this. But it seems like there should be one. Can anyone help me simplify this?
If it helps clarify here is a drawing of what I have tried to describe above:
The equation containing $csc$ should not have it at all, else ok.
$ x = \arctan { \sqrt{ \tan ^2 \phi + \tan ^2 \theta .}} $