tan + tan - = 2, ( 0 90^ ), then the value of is?

Answer: Option B

$$\begin{array}{rl}& \frac{\text{tan}\theta +\cot \theta }{\text{tan}\theta -\cot \theta }=2\\ & \text{By componendo and dividendo}\\ & \Rightarrow \frac{2\text{tan}\theta }{2\text{cos}\theta }=\frac{3}{1}\\ & \Rightarrow \frac{\sin \theta }{\text{cos}\theta }\times \frac{\sin \theta }{\text{cos}\theta }=3\\ & \Rightarrow {\sin }^2\theta =3\text{co}{\text{s}}^2\theta \\ & \Rightarrow {\sin }^2\theta =3\left(1-{\sin }^2\theta \right)\\ & \Rightarrow 4{\sin }^2\theta =3\\ & \Rightarrow {\sin }^2\theta \Rightarrow \frac{3}{4}\\ & \Rightarrow \text{sin }\theta =\frac{\sqrt{3}}{2}\\ & \\ & \mathbf{Alternate}:\\ & \Rightarrow \frac{\text{tan}\theta +\cot \theta }{\text{tan}\theta -\cot \theta }=2\\ & \text{By C and D}\\ & \Rightarrow \frac{\text{tan}\theta }{\cot \theta }=\frac{3}{1}\\ & \Rightarrow \text{ta}{\text{n}}^2\theta =3\\ & \Rightarrow \text{tan}\theta =\sqrt{3}\\ & \theta =60^{\circ }\\ & \Rightarrow \sin \theta \\ & \Rightarrow \text{sin }60^{\circ }\\ & \Rightarrow \frac{\sqrt{3}}{2}\end{array}$$