If $1+\cos ^{2} \theta=3 \sin \theta \cos \theta$, then the integral value of $\cot \theta\left(0<\theta<\frac{\pi}{2}\right)$ is

1 + cos²θ = 3 sinθ . cosθ Dividing both sides by sin²θ $\frac{1}{\sin ^{2} \theta}+\frac{\cos ^{2} \theta}{\sin ^{2} \theta}=\frac{3 \sin \theta \cos \theta}{\sin ^{2} \theta}$ $\Rightarrow \operatorname{cosec}^{2} \theta+\cot ^{2} \theta=3 \cot \theta$ $\Rightarrow 1+\cot ^{2} \theta+\cot ^{2} \theta=3 \cot \theta$ $\Rightarrow 2 \cot ^{2} \theta-3 \cot \theta+1=0$ $\Rightarrow 2 \cot ^{2} \theta-2 \cot \theta-\cot \theta+1=0$ $\Rightarrow 2 \cot ^{2} \theta(\cot \theta-1)-1(\cot \theta-1)=0$ $\Rightarrow(2 \cot \theta-1)(\cot \theta-1)=0$ $\Rightarrow \cot \theta=\frac{1}{2}$ or 1

Alternative:

Let $\theta=45^{\circ}$

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

$1+\left(\frac{1}{\sqrt{2}}\right)^{2}=3 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}$

$\frac{3}{2}=\frac{3}{2} \quad$ (Satisfies)

Now, $\cot \theta=\cot 45^{\prime}=1$