Question

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

Answer 1

Answer: 16

$7{\cos }^2\theta -3{\sin }^2\theta -2{\cos }^{2}2\theta =2$
$4{\cos }^2\theta +3\cos 2\theta -2{\cos }^{2}2\theta =2$
$2(1+\cos 2\theta )+3\cos 2\theta -2{\cos }^{2}2\theta =2$
$2{\cos }^{2}2\theta -5\cos 2\theta =0$
$\cos 2\theta (2\cos 2\theta -5)=0$
$\cos 2\theta =0$
$2\theta =(2n+1)\frac{\pi }{2}$
$\theta =(2n+1)\frac{\pi }{4}$
$S=\left\{\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}\right\}$
For all four values of $\theta$
$x^2-2\left({\tan }^2\theta +{\cot }^2\theta \right)x+6{\sin }^2\theta =0$
$\Rightarrow x^2-4x+3=0$
Sum of roots of all four equations $=4\times 4=16$.