Q) If 1 + sin2 θ = 3 sin θ cos θ, then prove that tan θ = 1 or 1/2
Ans:
Given that 1 + sin2 θ = 3 sin θ cos θ
We know that sin2 θ + cos2 θ = 1
Hence, 1 + sin2 θ = 3 sin θ cos θ will become:
(sin2 θ + cos2 θ) + sin2 θ = 3 sin θ cos θ
2 sin2 θ + cos2 θ = 3 sin θ cos θ
Let’s divide both sides by cos2 θ, we get:
2 tan2 θ + 1 = 3 tan θ
2 tan2 θ – 3 tan θ + 1 = 0
Let’s consider tan θ = X,
then the above equation will become 2 X2 – 3 X + 1 = 0
By solving this equation for its roots, we get:
2 X2 – 2X – X + 1 = 0
2 X (X – 1) – (X – 1) = 0
(2 X – 1) (X – 1) = 0
Therefore X = 1 or 
Since we had taken X = tan θ,
Therefore, we get tan θ = 1 and tan θ =