If √3tanθ = 3sinθ, then what is the value of sin2θ - cos2θ?

Answer: Option B

$$\begin{array}{rl}& \sqrt{3}\tan \theta =3\sin \theta \\ & \Rightarrow \frac{\tan \theta }{\sin \theta }=\frac{3}{\sqrt{3}}\\ & \Rightarrow \frac{\sin \theta }{\cos \theta }\times \frac{1}{\sin \theta }=\frac{3}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}\\ & \Rightarrow \frac{1}{\cos \theta }=\sqrt{3}\\ & \Rightarrow \cos \theta =\frac{1}{\sqrt{3}}\\ & \Rightarrow {\cos }^2\theta =\frac{1}{3}\\ & \therefore {\sin }^2\theta -{\cos }^2\theta \\ & =1-{\cos }^2\theta -{\cos }^2\theta \\ & =1-2{\cos }^2\theta \\ & =1-2\times \frac{1}{3}\\ & =\frac{1}{3}\end{array}$$