Chapter 14.4, Problem 38HP

To determine

ToProve:The value of $\tan \frac{\theta }{2}=\frac{\sin \theta }{1+\cos \theta }$

Answer to Problem 38HP

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

Explanation of Solution

Given information:

We know that, the given trigonometric expression is $\tan \frac{\theta }{2}=\frac{\sin \theta }{1+\cos \theta }$

Calculation:

  $\begin{array}{l}In\Delta ABP:-\\ \tan \frac{\theta }{2}=\frac{AP}{BP}\\ \tan \frac{\theta }{2}=\frac{AP}{OB+OA}\\ In\Delta OAP:-\\ \sin \theta =\frac{AP}{OP}=\frac{AP}{1}=AP\\ \cos \theta =\frac{OA}{OP}=\frac{OA}{1}=OA\\ \therefore \tan \frac{\theta }{2}=\frac{AP}{OB+OA}\\ \tan \frac{\theta }{2}=\frac{\sin \theta }{1+\cos \theta }\end{array}$