If $$\sqrt 3 $$ tanθ = 3sinθ, then the value of (sin²θ - cos²θ) is?

Solution(By Examveda Team)

$$\eqalign{ & {\text{ }}\sqrt 3 \tan \theta = 3\sin \theta \cr & {\bf{Shortcut method:}} \cr & \Rightarrow {\text{ }}\sqrt 3 \frac{{\sin \theta }}{{\cos \theta }} = 3\sin \theta \cr & \Rightarrow \frac{{\sqrt 3 }}{{\cos \theta }} = 3 \cr & \Rightarrow \cos \theta = \frac{{\sqrt 3 }}{3} \cr & {\text{then perpendicular}} = \sqrt 6 \cr} $$

$$\eqalign{ & \Rightarrow \left( {{{\sin }^2}\theta - {\text{co}}{{\text{s}}^2}\theta } \right) \cr & \Rightarrow {\left( {\frac{P}{H}} \right)^2} - {\left( {\frac{B}{H}} \right)^2} \cr & \Rightarrow {\left( {\frac{{\sqrt 6 }}{3}} \right)^2} - {\left( {\frac{{\sqrt 3 }}{3}} \right)^2} \cr & \Rightarrow \frac{6}{9} - \frac{3}{9} \cr & \Rightarrow \frac{1}{3} \cr} $$