Derivation of the orbital analysis equation and its physical significance

[Hamiltonian]

TL;DR Summary

unable to prove/derive theoretically the relation between bond angle and s and p character of a hybridized orbital.

$$cos\theta = \frac {s}{1-s} = \frac{p-1}{p}$$
in this equation $\theta$ is the bond angle and $s$ and $p$ are the fractional s-character of the orbital and p-character of the orbital.
This is equation is used rigorously in showing that the s-character of the axial orbitals in a $sp^3d$ hybridized orbital is 0 and hence it is also used to prove bents rule which states that the most electronegative element in a $sp^3d$ hybridized orbital takes the axial positions of the trigonal bipyramid that is formed. But I am unable to find a formal proof/derivation of this equation on the internet or in my textbook.
Is this relation between the bond angle and the s/p character very obvious that it does not need a proof/derivation?
(also I am assuming there might be an experimental proof for this formula but I am not looking for that rather a theoretical proof of this equation will be nice)

 

[Borek]

Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.

 

[Hamiltonian]

Can't say this relation makes sense, for s>0.5 cos(θ) would be larger than 1.

I am so sorry I made a typo it is actually $$cos\theta = \frac{s}{s-1}$$

 

[Borek]

Still fails, now for s>0.5.

 

[Hamiltonian]

Still fails, now for s>0.5.

I thought it was clear no hybridized orbital can have an s character more than 50%(0.5 fractional) as sp hybridized orbital has the least number of orbitals(2) combining and has 50% s-character in both orbitals. Hence the equation does not fail.
also, this very fact was used in Dragos rule...

 

[Borek]

OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.

 

[Hamiltonian]

OK, for "standard", non fractional hybridizations it will hold.

Apparently I know a bit too much to properly understand the problem as defined here.

maybe this particular equation isn't very popular(because it isn't very general?)

 

[Borek]

I don't remember being taught it. Doesn't mean much.

On the second thought I could get s and p reversed. My bad.