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

Solution (By Examveda Team)

$$\eqalign{ & \sqrt 3 \tan \theta = 3\sin \theta \cr & \Rightarrow \frac{{\tan \theta }}{{\sin \theta }} = \frac{3}{{\sqrt 3 }} \cr & \Rightarrow \frac{{\sin \theta }}{{\cos \theta }} \times \frac{1}{{\sin \theta }} = \frac{3}{{\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }} \cr & \Rightarrow \frac{1}{{\cos \theta }} = \sqrt 3 \cr & \Rightarrow \cos \theta = \frac{1}{{\sqrt 3 }} \cr & \Rightarrow {\cos ^2}\theta = \frac{1}{3} \cr & \therefore \,{\sin ^2}\theta - {\cos ^2}\theta \cr & = 1 - {\cos ^2}\theta - {\cos ^2}\theta \cr & = 1 - 2{\cos ^2}\theta \cr & = 1 - 2 \times \frac{1}{3} \cr & = \frac{1}{3} \cr} $$