The value of $\int\limits_{\pi/4}^{\pi/2} \cot \theta \cdot \text{cosec}^2 \theta \, d\theta$ is:

$\frac{1}{2}$

The correct answer is Option (1) → $\frac{1}{2}$

Since, $I = \int\limits_{\pi/4}^{\pi/2} \cot \theta \cdot \text{cosec}^2 \theta \, d\theta$

Let $\cot \theta = t ⇒-\text{cosec}^2 \theta \, d\theta = dt$

When $\theta = \frac{\pi}{4}, t = 1$;

When $\theta = \frac{\pi}{2}, t = 0$

So, $I = \int\limits_{1}^{0} t(-dt)$

$I= \int\limits_{0}^{1} t \, dt = \left[ \frac{t^2}{2} \right]_{0}^{1} = \frac{1}{2}$