If √3tanθ = 3sinθ, find the value of sin2θ - cos2θ

1.	If √3tanθ = 3sinθ, find the value of sin2θ - cos2θ
Answer» √3tanθ = 3sinθ √3\(\frac{sinθ}{cosθ}\) = √3 x √3sinθ \(\frac{1}{cosθ}\) = √3 cosθ = \(\frac{1}{\sqrt{3}}\) Now, sin²θ - cos²θ = 1 - cos²θ - cos²θ = 1 - 2 cos²θ = 1 - 2 x \(\Big(\frac{1}{\sqrt{3}}\Big)^2\) = 1 - 2 x \(\frac{1}{3}=\frac{1}{3}\)

Answer»

√3tanθ = 3sinθ

√3\(\frac{sinθ}{cosθ}\) = √3 x √3sinθ

\(\frac{1}{cosθ}\) = √3

cosθ = \(\frac{1}{\sqrt{3}}\)

Now, sin²θ - cos²θ = 1 - cos²θ - cos²θ

= 1 - 2 cos²θ

= 1 - 2 x \(\Big(\frac{1}{\sqrt{3}}\Big)^2\)

= 1 - 2 x \(\frac{1}{3}=\frac{1}{3}\)

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