Let S = (0,2) : 7^2 - 3^2 - 2^22 = 2.

Question

Let $S = \left\{	heta\in(0,2\pi) : 7\cos^2	heta - 3\sin^2	heta - 2\cos^22	heta = 2ight\}$. Then the sum of roots of all the equations $x^2 - 2(	an^2	heta+\cot^2	heta)x + 6\sin^2	heta = 0$, $	heta\in S$, is \_\_\_\_\_\_\_\_\_\_\_

Answer: 16

Accepted Answer

$$7\cos^2\theta-3\sin^2\theta-2\cos^22\theta = 2 \implies 4\cdot\frac{1+\cos2\theta}2+3\cos2\theta-2\cos^22\theta = 2$$

$$\implies 2+5\cos2\theta-2\cos^22\theta=2 \implies \cos2\theta(5-2\cos2\theta)=0 \implies \cos2\theta=0 \ (\text{since } 5-2\cos2\theta\ne0)$$

$$\cos2\theta=0 = \frac{1-\tan^2\theta}{1+\tan^2\theta} \implies \tan^2\theta=1$$

Sum of roots of each quadratic is $2(\tan^2\theta+\cot^2\theta) = 2(1+1)=4$. Since $\tan\theta=\pm1$ gives $\theta=\dfrac\pi4,\dfrac{3\pi}4,\dfrac{5\pi}4,\dfrac{7\pi}4$ in $(0,2\pi)$ --- four distinct equations --- the total sum of all roots is $4\times4=16$.

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