Question

Let $S = \\{θ ∈ (0, 2π) : 7 cos^2θ – 3 sin^2θ – 2 cos^22θ = 2\\}$.
Then, the sum of roots of all the equations $x^2 – 2 (tan^2θ + cot^2θ) x + 6 sin^2θ = 0, θ ∈ S$, is _______.

Answer 1

7 cos2θ – 3 sin2θ – 2 cos22θ = 2
$\begin{array}{l} \Rightarrow 4\left(\frac{1+\cos2\theta}{2}\right)+3\cos2\theta-2\cos^22\theta=2 \end{array}$
⇒ 2 + 5 cos2θ – 2 cos2 2θ = 2
⇒ cos 2θ = 0 or 5/2 (rejected)
$\begin{array}{l} \Rightarrow \cos2\theta=0=\frac{1-\tan^2\theta}{1+\tan^2\theta}\Rightarrow \tan^2\theta =1\end{array}$
∴ Sum of roots = 2 (tan2θ + cot2θ) = 2 × 2 = 4
But as tanθ = ±1 for $π/4, 3π/4, 5π/4, 7π/4$ in the interval ($0, 2π$)
∴ Four equations will be formed
Hence, sum of roots of all the equations = 4 × 4 = 16.