The value of ${\sec }^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}\sec \left(\frac{7\pi }{12}+\frac{k\pi }{2}\right)\sec \left(\frac{7\pi }{12}+\frac{\left(k+1\right)\pi }{2}\right)\right)$ In the interval $\left[-\frac{\pi }{4},\frac{3\pi }{4}\right]$ is 

$\begin{array}{l}S=\sum _{k=0}^{10}\sec \left(\frac{7\pi }{12}+\frac{k\pi }{2}\right)\sec \left(\frac{7\pi }{12}+\frac{k\pi }{2}+\frac{\pi }{2}\right)\\ =-\sum _{k=0}^{10}2\cos ec\left(\frac{7\pi }{6}+k\pi \right)\\ =-2\sum _{k=0}^{10}(-1{)}^{k+1}\cos ec\left(\frac{\pi }{6}\right)\\ =-4(-1)=4\\ \therefore {\sec }^{-1}\left(\frac{4}{4}\right)={\sec }^{-1}\left(1\right)=0\end{array}$