$Assumptions = x ∈ Reals ;

(*(1+x^2)^(1/4)*) Cos[-(1/2) ArcTan[ x ] ] // Simplify

(*((TrigToExp[Cos[((ArcTan[x])/2)]] // Simplify) + I(TrigToExp[-Sin[((ArcTan[x])/2)]]  // Simplify) // Simplify)^2*)

ClearAll[t,f,sigma, epsilon, omega, mu, epsilon, alpha, beta]
t = sigma/(epsilon omega) ;
f= Sqrt[ omega^2 mu epsilon ] ;
beta = f Sqrt[ 1 + Sqrt[ 1 + t^2]]/Sqrt[2] 
alpha=  f t/(Sqrt[2]Sqrt[ 1 + Sqrt[ 1 + t^2]]) 

(alpha +  I beta)^2 // FullSimplify

ClearAll[format]

(*format = {{omega -> ω}, {sigma -> (Subscript[σ, 0])}, {mu -> μ}, {epsilon -> ϵ'}} ;*)

(((alpha /. omega -> ω) /. sigma -> (Subscript[σ, 0])) /. mu -> μ )/. epsilon -> ϵ'