Let $$S=\left\{\theta \in(0,2 \pi): 7 \cos ^{2} \theta-3 \sin ^{2} \theta-2 \cos ^{2} 2 \theta=2\right\}$$. Then, the sum of roots of all the equations $$x^{2}-2\left(\tan ^{2} \theta+\cot ^{2} \theta\right) x+6 \sin ^{2} \theta=0, \theta \in S$$, is __________.

<p>$7{\cos ^2}\theta - 3{\sin ^2}\theta - 2{\cos ^2}2\theta = 2$</p> <p>$$ \Rightarrow 4\left( {{{1 + \cos 2\theta } \over 2}} \right) + 3\cos 2\theta - 2{\cos ^2}2\theta = 2$$</p> <p>$\Rightarrow 2 + 5{\cos ^2}\theta - 2{\cos ^2}2\theta = 2$</p> <p>$\Rightarrow \cos 2\theta = 0$ or ${5 \over 2}$(rejected)</p> <p>$$ \Rightarrow \cos 2\theta = 0 = {{1 - {{\tan }^2}\theta } \over {1 + {{\tan }^2}\theta }} \Rightarrow {\tan ^2}\theta = 1$$</p> <p>$\therefore$ Sum of roots $= 2({\tan ^2}\theta + {\cot ^2}\theta ) = 2 \times 2 = 4$</p> <p>But as $\tan \theta = \, \pm \,1$ for ${\pi \over 4},{{3\pi } \over 4},\,{{5\pi } \over 4},\,{{7\pi } \over 4}$ in the interval $(0,\,2\pi )$</p> <p>$\therefore$ Four equations will be formed</p> <p>Hence sum of roots of all the equations $= 4 \times 4 = 16.$</p>