Answer
$\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta=\dfrac{3-\sqrt{3}}{3}$
Work Step by Step
$\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta$
Integrate the expression directly and apply the second part of the fundamental theorem of calculus to get the answer:
$\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta=-\cot\theta\Big|_{\pi/4}^{\pi/3}=-\cot\dfrac{\pi}{3}-(-\cot\dfrac{\pi}{4})=-\dfrac{\sqrt{3}}{3}+1=\dfrac{3-\sqrt{3}}{3}$