Question

$if \: 1 + {sin}^{2}\theta \: = 3sin\theta\times cos \theta \: \: \\ then \: prove \: that \: tan \theta \: = 1 \: or \: tan \theta = \frac{1}{2}$
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Answer 1

Given: 1+sin2 θ = 3 sin θ cos θ Dividing L.H.S and R.H.S equations with sin2 θ, We get, cosec2 θ + 1 = 3 cot θ Since, cosec2 θ – cot2 θ = 1 ⇒ cosec2 θ = cot2 θ +1 ⇒ cot2 θ +1+1 = 3 cot θ ⇒ cot2 θ +2 = 3 cot θ ⇒ cot2 θ –3 cot θ +2 = 0 Splitting the middle term and then solving the equation, ⇒ cot2 θ – cot θ –2 cot θ +2 = 0 ⇒ cot θ(cot θ -1)–2(cot θ +1) = 0 ⇒ (cot θ – 1)(cot θ – 2) = 0 ⇒ cot θ = 1, 2 Since, tan θ = 1/cot θ tan θ = 1, ½ Hence, proved.Read more on Sarthaks.com - https://www.sarthaks.com/884852/if-1-sin-2-3sin-cos-then-prove-that-tan-1-or