Question : If $1 + \sin^2 θ - 3\sinθ \cosθ = 0$, then the value of $\cotθ$ is:
Option 1: $0$
Option 2: $2$
Option 3: $\frac{1}{2}$
Option 4: $\frac{1}{3}$
Correct Answer: $2$
Solution :
Given: $1 + \sin^2 θ - 3\sinθ \cosθ = 0$
⇒ $\sin^2 θ + \cos^2 θ+ \sin^2 θ - 2\sinθ \cosθ- \sinθ \cosθ = 0$ [As $\sin^2 θ + \cos^2 θ = 1$]
⇒ $\sin^2 θ + \cos^2 θ- 2\sinθ \cosθ = \sinθ \cosθ- \sin^2 θ$
⇒ $(\sin θ - \cos θ)^2= -\sinθ(- \cosθ+ \sin θ)$
⇒ $(\sin θ - \cos θ)(\sin θ - \cos θ)= -\sinθ(\sin θ-\cosθ)$
⇒ $(\sin θ - \cos θ)= -\sinθ$
⇒ $2\sin θ =\cos θ$
⇒ $2=\cot\theta$ [dividing by $\sin θ$ in both sides]
$\therefore\cot\theta = 2$
Hence, the correct answer is $2$.
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