Question 3.7: Show that equation E′ = 1/2(1 − α)E, which frequently arises......

In the equation E′ = 1/2(1 − α)E, the value of α is given by α = (A – 1)^2/(A + 1)^2. The value of (1 − α) is therefore [(A + 1)^2 − (A − 1)^2/(A + 1)^2 = [A^2 + 2A + 1 – A^2 + 2A − 1]/(A + 1)^2 = 4A/(A + 1)^2. The value of E′ is therefore E^′ = 2AE/(A + 1)^2, which is equivalent to Equation 3.91.

\mathbf{E}^{\prime}={\frac{2\mathbf{A}\mathbf{E}}{\left(\mathbf{A}+1\right)^{2}}}         (3.91)