\(F+F \cos \theta=m g \sin \theta\)

\(F=\frac{m g \sin \theta}{1+\cos \theta}\)

\(F=\frac{m g 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^2 \frac{\theta}{2}}\) \(\left(\because \sin \theta=2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}\right.\) and \(\left.1+\cos \theta=2 \cos ^2 \frac{\theta}{2}\right)\)

\(=m g \tan \frac{\theta}{2}\)