Question

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

Answer 1

Step 1: Simplify the summation

Let $S=$$se{c}^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}sec\left(\frac{7\mathrm{\pi }}{12}+\frac{k\mathrm{\pi }}{2}\right)sec\left(\frac{7\mathrm{\pi }}{12}+\frac{\left(k+1\right)\mathrm{\pi }}{2}\right)\right)$

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}sec\left(\frac{7\mathrm{\pi }}{12}+\frac{k\mathrm{\pi }}{2}\right)sec\left(\frac{7\mathrm{\pi }}{12}+\frac{\mathrm{k\pi }}{2}+\frac{\mathrm{\pi }}{2}\right)\right)$

Step 2: Apply complementary trigonometric identity and simplify the summation

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{4}\sum _{k=0}^{10}sec\left(\frac{7\mathrm{\pi }}{12}+\frac{k\mathrm{\pi }}{2}\right)cosec\left(\frac{7\mathrm{\pi }}{12}+\frac{\mathrm{k\pi }}{2}\right)\right)$ $...\left[\because sec\left(90+\theta \right)=-\cos ec\theta \right]$

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{4}\sum _{k=0}^{10}\frac{1}{\cos \left({\displaystyle \frac{7\mathrm{\pi }}{12}}+{\displaystyle \frac{\mathrm{k\pi }}{2}}\right)\sin \left({\displaystyle \frac{7\mathrm{\pi }}{12}}+{\displaystyle \frac{k\mathrm{\pi }}{2}}\right)}\right)$

Step 3: Apply multiple angle trigonometric identity and simplify the summation

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{4}\sum _{k=0}^{10}\frac{2}{2\sin \left({\displaystyle \frac{7\mathrm{\pi }}{12}}+{\displaystyle \frac{\mathrm{k\pi }}{2}}\right)\cos \left({\displaystyle \frac{7\mathrm{\pi }}{12}}+{\displaystyle \frac{k\mathrm{\pi }}{2}}\right)}\right)$ $...\left[\because 2\sin \theta \cos \theta =\sin 2\theta \right]$

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{2}\sum _{k=0}^{10}\frac{1}{\sin \left({\displaystyle \frac{7\mathrm{\pi }}{6}}+{\displaystyle k\mathrm{\pi }}\right)}\right)$

Step 4: Use the general solution of trigonometric equation

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{2}\sum _{k=0}^{10}\frac{1}{\sin \left({\displaystyle \left(k+\right)1\mathrm{\pi }+\frac{\mathrm{\pi }}{6}}\right)}\right)$ $...\left[\because \sin \left(n\mathrm{\pi }+\mathrm{\theta }\right)={\left(-1\right)}^n\sin \theta \right]$

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{2}\sum _{k=0}^{10}\frac{1}{{\left(-1\right)}^{k+1}\sin \left({\displaystyle \frac{\mathrm{\pi }}{6}}\right)}\right)$ $...\left[\because \sin \frac{\mathrm{\pi }}{6}=\frac{1}{2}\right]$

$\Rightarrow$ $S=$$se{c}^{-1}\left(\frac{-1}{2}\sum _{k=0}^{10}\frac{1}{{\left(-1\right)}^{k+1}\times {\displaystyle \frac{1}{2}}}\right)$

$\Rightarrow$ $S=$$se{c}^{-1}\left(-\sum _{k=0}^{10}\frac{1}{{\left(-1\right)}^{k+1}}\right)$

Step 5: Expand the summation

$\Rightarrow$ $S=$$se{c}^{-1}\left(-\left(-1+1-1+....-1\right)\right)$

$\Rightarrow$ $S=$$se{c}^{-1}\left(1\right)$

$\Rightarrow$ $S=0$

Hence, the value of the expression $se{c}^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}sec\left(\frac{7\mathrm{\pi }}{12}+\frac{k\mathrm{\pi }}{2}\right)sec\left(\frac{7\mathrm{\pi }}{12}+\frac{\left(k+1\right)\mathrm{\pi }}{2}\right)\right)$ is $0$.