This question is taken from RS Aggarwal book, Trigonometry, topic – trigonometric identities and we have to prove that tanθ = 1 or 1/2
and we have given 1+sin²θ = 3sinθcosθ
Class 10, chapter 13A, question no 38.
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1+sin²θ = 3sinθcosθ
Divide by cos²θ both sides, we get
1/cos²θ + sin²θ/cos²θ = 3sinθ/cosθ
sec²θ+tan²θ = 3tanθ
1+tan²θ+tan²θ = 3tanθ ( By Identity — sec²θ = 1+tan²θ)
1+2tan²θ = 3tanθ
Now let tanθ=a
1+2a²=3a
or 2a²−3a+1=0
(2a−1)(a−1)=0
By solving, we get
a=1/2or1
Thus, tanθ=1or1/2