If tan ( /2 - /2 ) = 3 the value of cosθ is?

Answer: Option C

$$\begin{array}{rl}& \text{ tan}\left(\frac{\pi }{2}-\frac{\theta }{2}\right)=\sqrt{3}\\ & \Rightarrow \text{tan}\left({90}^{\circ }-\frac{\theta }{2}\right)=\sqrt{3}\left[\pi ={180}^{\circ }\right]\\ & \Rightarrow \cot \frac{\theta }{2}=\sqrt{3}\\ & \Rightarrow \cot \frac{\theta }{2}=\cot 30^{\circ }\\ & \Rightarrow \frac{\theta }{2}=30^{\circ }\\ & \Rightarrow \theta =60^{\circ }\\ & \Rightarrow \cos 60^{\circ }=\frac{1}{2}\end{array}$$