$\begin{aligned} I_n & =\int_{\pi / 4}^{\pi / 2} \cot ^n x d x=\int_{\pi / 4}^{\pi / 2} \cot ^{n-2} x\left(\operatorname{cosec}^2 x-1\right) d x \\ & \left.=-\frac{\cot ^{n-1} x}{n-1}\right]_{\pi / 4}^{\pi / 2}-I_{n-2} \\ & =\frac{1}{n-1}-I_{n-2} \\ & \Rightarrow \quad I_n+I_{n-2}=\frac{1}{n-1} \\ & \Rightarrow \quad I_2+I_4=\frac{1}{3} \\ & I_3+I_5=\frac{1}{4} \\ & I_4+I_6=\frac{1}{5} \\ & \quad \frac{1}{I_2+I_4}, \frac{1}{I_3+I_5}, \frac{1}{I_4+I_6} \text { are in A.P. }\end{aligned}$