Considering Section 7.1, suppose we began the analysis to find E = E₁+ E₂with two cosine functions E₁= E₀₁cos (ωt + α₁) and E₂= E₀₂ cos (ωt + α₂). To make things a little less complicated, let E₀₁= E₀₂ and α₁ = 0. Add the two waves algebraically and make use of the familiar trigonometric identity cos θ + cos Φ = 2 cos 1/2 (θ + Φ) cos 1/2 (θ + Φ) in order to show that E = E₀cos (ωt + α), where E₀= 2E₀₁cos α₂/2 and α = α₂/2. Now show that these same results follow from Eqs. (7.9) and (7.10).

E = Ei + E2 + 2E01E02 cos (2 aj) (7.9) E01 sin aj + E2 sina2 tan a (7.10) E01 cos aj + E02 cos a2