The arithmetic mean of the roots of the equation

$4\cos ^{ 2 }{ x } -4\cos ^{ 2 }{ x } -\cos { (315\pi +x) } =1$ in the interval $(0,315)$ is

Given

$4\cos ^{ 3 }{ x } -4\cos ^{ 3 }{ x } -\cos { \left( 315\pi +x \right) } =1\quad \quad$

$\Rightarrow 4\cos ^{ 3 }{ x } -4\cos ^{ 3 }{ x } -\cos { x } -1=0\quad \left[ \because \cos { \left( 315\pi +x \right) } ={ (-1) }^{ 315 }\cos { x } =-\cos { x } \right]$

$\Rightarrow \left( 4\cos ^{ 2 }{ x } +1 \right) \left( \cos { x } -1 \right) =0$

$\Rightarrow \cos { x } -1=0$

$4\cos ^{ 2 }{ x } +1\neq 0$

$\Rightarrow \cos { x } =1\quad$

$\Rightarrow \cos { x } =\cos { 0 } \quad$

$\Rightarrow x=2n\pi ,n\in I$

$\quad \therefore x=2\pi ,4\pi ,6\pi ,8\pi ,...100\pi \left[ \because 0<x<315 \right]$

$\therefore$ Required Arithmetic mean

$=\cfrac { 2\pi +4\pi +6\pi +8\pi ,...+100\pi }{ 50 } =\cfrac { 2\pi .\cfrac { 50 }{ 2 } .51 }{ 50 } =51\pi$