Suppose we began the analysis to find E=E₁+E₂ with two cosine functions E₁=E₀₁cos(ωₜ+α) and E₂=E₀₂cos(ωₜ+α₂). To make things a little less complicated, let E₀₁=E₀₂ and αn=0. Add the two waves algebraically and make use of the familiar trigonometric identity cos(θ)+cos(φ)=2cos1/2(θ+φ)cos1/2(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):
E₀²=E₀₁²+E₀₂²+E₀₁E₀₂cos(α₂−α₁) (7.9)
That's the sought-after expression for the amplitude (Ev) of the resultant wave. Now to get the phase, divide Eq. (7.8) by (7.7):
E₀₁sinα+E₀₂sinα₂/tanα=E01cosα+E₀₂cosα₂ (7.10)
