Rearrange equation to isolate for P to use in a calculator please
v=e[(1/cos(pi/2sqrt(P/Pcr)))-1] OR
e[(sec(pi/2sqrt(P/Pcr)-1]
v=e[(1/cos(π/2√(P/Pcr)))-1]
v=e/cos(π/2√(P/Pcr)) - e
v+e = e/cos(π/2√(P/Pcr))
cos(π/2√(P/Pcr)) = e/(v+e)
π/2√(P/Pcr) = arccos(e/(v+e))
Now, assuming the usual carelessness with parentheses,
π/arccos(e/(v+e)) = 2√(P/Pcr)
(π / 2arccos(e/(v+e)))^2 = P/Pcr
P = Pcr (π / 2arccos(e/(v+e)))^2
assuming Pcr is a single quantity. Otherwise, the P's cancel!