Chapter 7.4, Problem 37E

To determine

To prove: the given equation.

Answer to Problem 37E

See explanation

Explanation of Solution

Given information:

The figure representing the unit circle $O$ is shown below.

  

  $\tan \frac{1}{2}\theta =\frac{\sin \theta }{1+\cos \theta }$

Calculation:

Since $\angle PBD$ subtends the same arc as the central $\angle POD$ , then it has the half measure of $\angle POD$ , i.e.

  $\angle PBD=\frac{1}{2}\angle POD=\frac{1}{2}\theta$

Use the right $\Delta PBA$ as shown below.

  $\begin{array}{c}\tan \frac{1}{2}\theta =\frac{PA}{BA}\\ \tan \frac{1}{2}\theta =\frac{PA}{BO+OA}\\ \tan \frac{1}{2}\theta =\frac{\sin \theta }{1+\cos \theta }\end{array}$

Thus, the given equation is proved.