Is my trig result unique? I recently determined that for all integers $a$ and $b$ such that $a\neq b$ and $b\neq 0$,
$$
\arctan\left(\frac{a}{b}\right) + \frac{\pi}{4} = \arctan\left(\frac{b+a}{b-a}\right)
$$
This implies that 45 degrees away from any angle with a rational value for tangent lies another angle with a rational value for tangent.  The tangent values are related.
If anyone can let me know if this has been done/shown/proven before, please let me know.  Thanks!

Quoting from Wikipedia's list of trigonometric identities:
BEGIN QUOTE
$$ f(x) = \frac{(\cos\alpha)x - \sin\alpha}{(\sin\alpha)x + \cos\alpha}, $$
[$\ldots\ldots$ some material omitted here $\ldots\ldots$]
If $x$ is the slope of a line, then $f(x)$ is the slope of its rotation through an angle of $-\alpha$.
END QUOTE
Dividing the numerator and denominator by $\tan\alpha$ may give the same result as is posted here.
