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 =