Question

Let $S=\left\{\theta \in(0,2 \pi): 7 \cos ^2 \theta-3 \sin ^2 \theta-2\right.$ $\left.\cos ^2 2 \theta=2\right\}$. 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

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