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Question

If 3 tan θ = 3sin θ then (sin2θ – cos2θ) = ?

(a) 13

(b) 13

(c) 3

(d) 23

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Solution

Given:3tanθ=3sinθ3tanθ=3sinθ3sinθcosθ=3sinθ3sinθcosθ-3sinθ=03sinθ-3sinθcosθcosθ=03sinθ-3sinθcosθ=03sinθ1-3cosθ=0sinθ=0 or 1-3cosθ=0sinθ=0 or cosθ=13sinθ=0 or cos2θ=13sinθ=0 or 1-cos2θ=1-13sinθ=0 or sin2θ=23sinθ=0 or sinθ=23For sinθ=0,sin2θ=01-sin2θ=1-0cos2θ=1Thus, sin2θ-cos2θ=-1For sinθ=23,sin2θ=231-sin2θ=1-23cos2θ=13Thus, sin2θ-cos2θ=23-13=13

Hence, the correct option is (a).

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