I have found an equation in the book of Robert Boylestad & Louis Nashelsky, as (27) on page 277 of the 2014 edition:
Which pertains to the circuit depicted in Fig.29-30 on page 275 of the book.
But that equation is not matching with the equation which one gave solve for this type of question previously.
$$Z_o=R_c \parallel \left[r_o+ \frac{\beta (r_o+r_e)}{1+\beta r_e/R_E}\right]$$
How has this equation been derived?
No equation is correct by 100% . Each equation contains simplifications/neglections.
Therefore, slight deviations between formulas are normal - and you should tell us which equations you are referring to.
As far as the output impedance of a common emitter stage is concerned - this value depends also on the kind of signal source at the base (with or without an internal source resistance?). This might be the reason for different expressions?
The equation you give matches perfectly equation (27) from the book that you presumably are referring to. However, there seems to be a problem with either that equation or the circuit drawn in which the 'output current' \$I_o\$ seems to be the current through the collector's resistor \$R_C\$.
Normally the output impedance refers to an output current (which would then be called \$I_o\$) which results from applying a voltage (\$V_o\$) to the output of the circuit while the input (the base) is shorted (connected to signal ground).
In the picture where \$Z_o\$ and \$V_o\$ are drawn where one would expect them, \$I_o\$ is given as the collector current. Later (p277) they (Boylestad and Nashelsky) argue that as \$R_E >> r_e\$, \$Z_o=R_C\$, which seems a bit obvious given their definition of \$I_o=I_C\$.
