Question

If 1 +sintetta =3sinteetta ×costetta prove that tantetta =1or 1/2

Answer 1

Given : 1 + sinθ = 3 sinθ cosθ

Dividing both sides by cos²θ, we get

⇒ $\dfrac{1}{cos \theta } + \dfrac{sin \theta }{cos^{2}\theta} = \dfrac{3 sin \theta cos \theta}{cos^{2} \theta}$

⇒ sec²θ + tan²θ = 3 tanθ

⇒ ( 1 + tan²θ ) + tan²θ = 3 tanθ

⇒ 1 + tan²θ + tan²θ = 3 tanθ

⇒ 1 + 2 tan²θ - 3 tanθ = 0

⇒ 2 tan²θ - 3 tanθ + 1 = 0

⇒ 2 tan²θ - ( 2 + 1 ) tanθ + 1 =  0

⇒ 2 tan²θ - 2 tanθ - tanθ + 1 = 0

⇒ 2 tanθ( tanθ - 1 ) - 1( tanθ - 1 ) = 0

⇒ ( tanθ - 1 )( 2 tanθ - 1 ) = 0

⇒ tanθ = 1 or tanθ = $\dfrac{1}{2}$

Hence, proved that if 1 + sinθ = 3sinθcosθ, then tanθ = 1 or 1 / 2

Answer 2

See in attachment
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Hope it will help you

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