Tell MATLAB to keep slimplfying until it cannot simplify it further (or reaches the iteration limit), then test equality with the isAlways function:
syms mu B cc M M_bar E V_N sigma_1;
sigma_1 = sqrt((M+M_bar-4*E*V_N)^2+(2*mu*B)^2);
cantSimplified_Var = simplify(sqrt(4*B^2*mu^sym(2) + 16*E^sym(2)*V_N^2 - 8*E*M*V_N - 8*E*M_bar*V_N + M^2 + 2*M*M_bar + M_bar^2), 'Steps',500);
Test = isAlways(sigma_1 == cantSimplified_Var)
producing:
Test =
logical
1
.
