The value of $\sec^{-1}\left(\frac{1}{4}\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12}+\frac{k\pi}{2}\right)\sec\left(\frac{7\pi}{12}+\frac{(k+1)\pi}{2}\right)\right)$ in the interval $\left[-\frac{\pi}{4},\ \frac{3\pi}{4}\right]$ equals ______.

The value of sec⁻¹(1/4∑ₖ₌₀¹⁰sec(7π/12+kπ/2)sec(7π/12+((k+1)π)/2)) in the interval [-π/4, 3π/4] equals ______. $\text{Given: } \sec^{-1}\!\left(\dfrac{1}{4}\sum_{k=0}^{10}\sec\!\left(\dfrac{7\pi}{12}+\dfrac{k\pi}{2}\right)\sec\!\left(\dfrac{7\pi}{12}+\dfrac{(k+1)\pi}{2}\right)\right)$ $\text{angle in } \left[-\dfrac{\pi}{4},\ \dfrac{3\pi}{4}\right]$ $\text{To find: the value of the expression in } \left[-\dfrac{\pi}{4},\ \dfrac{3\pi}{4}\right]$ $1.\ \text{relate the two angles in each term}$ $2.\ \text{simplify } \sec A \sec B$ $3.\ \text{sum the terms, then apply } \sec^{-1}$ $A = \dfrac{7\pi}{12} + \dfrac{k\pi}{2},\quad B = \dfrac{7\pi}{12} + \dfrac{(k+1)\pi}{2}$ $\Rightarrow B = A + \dfrac{\pi}{2}$ $\sec A \sec B = \dfrac{1}{\cos A \cos B}$ $\cos B = \cos\!\left(A+\dfrac{\pi}{2}\right) = -\sin A$ $\cos A \cos B = -\sin A \cos A = -\dfrac12\sin 2A$ $\sec A \sec B = \dfrac{1}{-\frac12 \sin 2A} = -\dfrac{2}{\sin 2A} = -2\,\operatorname{cosec} 2A$ $2A = \dfrac{7\pi}{6} + k\pi$ $\sin\!\left(\dfrac{7\pi}{6}+k\pi\right) = (-1)^k \sin\dfrac{7\pi}{6}$ $\sin\dfrac{7\pi}{6} = -\dfrac12$ $\sin\!\left(\dfrac{7\pi}{6}+k\pi\right) = \dfrac{(-1)^{k+1}}{2}$ $\sec A \sec B = -\dfrac{2}{(-1)^{k+1}/2} = 4(-1)^k$ $\sum_{k=0}^{10}\sec A \sec B = 4\sum_{k=0}^{10}(-1)^k$ $\sum_{k=0}^{10}(-1)^k = 1 \ \Rightarrow\ \sum = 4$ $\dfrac{1}{4}\sum = \dfrac{1}{4}\times 4 = 1$ $\sec^{-1}(1) = 0$ $\text{and } 0 \in \left[-\dfrac{\pi}{4},\ \dfrac{3\pi}{4}\right] \ \Rightarrow\ \text{valid}$ $\boxed{0}$