Question

$\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\_\_\_\_\_}$.

Short answer

$\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\underset{⏝}{\sin \theta }}$.

Step-by-step solution

Step 1. Given Data  

The given incomplete equation is

$\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\_\_\_\_\_}$

Step 2. Explanation

Use tangent formula

$$\begin{gathered} \tan \frac{\theta }{2}=\frac{\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}} \\ \tan \frac{\theta }{2}=\frac{2{\sin }^2\frac{\theta }{2}}{2\left(\sin \frac{\theta }{2}\right)\left(\cos \frac{\theta }{2}\right)} \\ \tan \frac{\theta }{2}=\frac{1-\cos \left(2{\displaystyle \frac{\theta }{2}}\right)}{\sin \left(2\frac{\theta }{2}\right)} \\ \tan \frac{\theta }{2}=\frac{1-\cos \theta }{\sin \theta } \end{gathered}$$