Theorem: 2 sin^2(pi/4) - 1 = 0 (TIR sqrt-2-medium 45-degree critical angle)

Theorem (TIR-pi-over-4 canonical): for n_1 = sqrt(2), n_2 = 1 the critical angle theta_c satisfies sin theta_c = 1/sqrt(2), so sin^2 theta_c = 1/2 and 2 sin^2 theta_c - 1 = 0 identically. Canonical sympy pin: residual =…