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Question 4
If 1+sin2 θ=3 sin θ.cos θ, then prove that tan θ=1 or 12.


Solution

Given, 1+sin2 θ=3 sin θ.cos θ

On dividing by sin2 θ on both sides, we get;

1sin2 θ+1=3.cot θ      [cot θ=cos θsin θ]

cosec2 θ+1=3.cot θ      [cosec θ=1sin θ]

1+cot2 θ+1=3.cot θ      cosec2 θcot2 θ=1

cot2 θ3 cot θ+2=0

cot2 θ2 cot θcot θ+2=0    [by splitting the middle term]

cot θ(cot θ2)1(cot θ2)=0

(cot θ2)(cot θ1)=0cot θ=1 or 2

tan θ=1 or 12     [tan θ=1cot θ]

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