Chapter 5, Problem 43E

Obtain expression (5-23) from equation (5-22). Using $\cos \theta ={\cos }^2\left(\frac{1}{2}\theta \right)-{\sin }^2\left(\frac{1}{2}\theta \right)$ and $\sin \theta =2\sin \left(\frac{1}{2}\theta \right)\cos \left(\frac{1}{2}\theta \right)$ , first convert the argument of the cotangent fromkL to $\frac{1}{2}kL$ . Next, put the resulting equation in quadratic form, and then factor. Note that $\alpha$ is positive by definition.