It would be nice if we could all gather at a table with pencil and paper. This tex formatting makes the discussion a little difficult. The derivation I gave is not my own; as I said I got it from my old textbook and therefore is most probably correct.It's not the same as in his original post. What he had in his original post is:
"is the expression Fb= 1/2\(\pi\)r(C\(_{}\pi\) + C\(_{}\mu\))...correct?"
Which may or may not be correct depending on what r he's referring to.
He posted in post #3 some pdf files (3.pdf, 4.pdf) that give this derivation:
\(f_\beta=\frac{f_\tau}{\beta_{mid}}=\frac{1}{2\pi r_\pi(C_\pi C\mu)}\)
so
\(f_\tau=\frac{1}{2\pi r_{e}(C_\pi C\mu)}\)
These expressions are equivalent to yours, but since the OP didn't specify which r he is referring to in his original post, we don't know which of your expressions might be equivalent to his. As his expression stands, none of yours are equivalent to his.
Also, he is somewhat confused over what Fb is. He said in post #1:
"the question is to find an expression for β cutoff frequency Fb which is simply a high freq current gain cutoff point of the transistor with collector & emitter terminals shorted as seen in the attached"
The file 4.pdf he attached says "The transit or cut-off frequency, Ft,...is defined as the frequency where the current gain falls to 1."
They've used the phrase "cutoff frequency" to mean "the frequency where the current gain falls to 1"
On the other hand, his reference material (3.pdf) does refer to a frequency \(f_\beta\), which is apparently the 3dB down frequency.
So, my best guess is that he wants:
\(f_\beta=\frac{1}{2\pi r_\pi(C_\pi C\mu)}\)
The schematic he posted has a resistor R∏ as a series element, which is not correct; R∏ is the shunt element. The series element is Rx in Sedra and Smith. Having R∏ as a series element has no effect on Ft when calculated for a transistor driven by a current source, but it will affect the response when driven from a more typical source.
Sedra and Smith don't include the shunt element RB, and the fact that the OP included it led me to think that perhaps his instructor intended it to be a part of the circuit for which Ft is to be calculated.