Chapter 7, Problem 63RE

To determine

To establish:the given identity.

Answer to Problem 63RE

The identity is established.

Explanation of Solution

Given:

The given identity is $\left(1+\cos \theta \right)\tan \frac{\theta }{2}=\sin \theta$ .

Calculation:

As the given identity is $\left(1+\cos \theta \right)\tan \frac{\theta }{2}=\sin \theta$ .

Now, let us check LHS:

  $\begin{array}{l}LHS=\left(1+\cos \theta \right)\tan \frac{\theta }{2}\\ \left\{\tan \frac{\theta }{2}=\left(\frac{\sin }{1+\cos \theta }\right)\right\}\\ =\left(1+\cos \theta \right)\left(\frac{\sin }{1+\cos \theta }\right)\\ =\sin \theta \\ =RHS\end{array}$

Thus, $\left(1+\cos \theta \right)\tan \frac{\theta }{2}=\sin \theta$

Hence, the identity is established.