Question : What is $\tan \frac{\theta}{2}$?
Option 1: $\frac{\cos \theta}{1-\sin \theta}$
Option 2: $\frac{\sin \theta}{1-\cos \theta}$
Option 3: $\frac{\cos \theta}{1-\cos \theta}$
Option 4: $\frac{\sin \theta}{1+\cos \theta}$
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Correct Answer: $\frac{\sin \theta}{1+\cos \theta}$
Solution :
Given: $\tan\frac{\theta}{2}$
= $\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}}$
= $\frac{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2\cos^2\frac{\theta}{2}}$
= $\frac{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2\cos^2\frac{\theta}{2}-1 + 1 }$
= $\frac{\sin\theta}{\cos\theta + 1}$
Hence, the correct answer is $\frac{\sin \theta}{\cos\theta + 1}$.
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