Question: It's best that I use a diagram to illustrate the question:

[![s](https://i.stack.imgur.com/GiuEd.jpg)](https://i.stack.imgur.com/GiuEd.jpg)

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> Is there any interesting relationship, one that can be expressed mathematically, between $\angle CDB$ and $\angle CAB$
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Answer: 
As your answer above states, $\angle CDB=180^\circ-\angle CAB$. To derive this, we can construct kite $ABDC$. $m\angle DCA$ and $m\angle DBA$ are both equal to $90^\circ$, being radii to points of tangency. Since circumscribed angle $\angle CDB$ is opposite $\angle CAB$ in kite $ABDC$ which are between pairs of congruent sides (i.e., AC = BA and CD = BD), $\angle CAB$ is supplementary to $\angle CDB$.

One interesting result of this is that the measure of the *major* arc $BC$ is equal to $180^\circ$ + $m\angle CDB$. 

This is because the measure of minor arc $BC$ is equal to $m\angle CDB$, $360^\circ$ - $m\angle CDB$ gives the measure of major arc $BC$. These relationships hold true for any circumscribed angle of any circle.
Unfortunately, this is rather elementary. And it's just actually:
$$\angle CDB=180^\circ-\angle CAB$$