Let S=left{theta in(0,2 pi): 7 cos ^{2} theta | vedclass

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Question

$7 \cos ^{2} 	heta-3 \sin ^{2} 	heta-2 \cos ^{2} 2 	heta=2$

$4 \cos ^{2} 	heta+3 \cos 2 	heta-2 \cos ^{2} 2 	heta=2$

$2(1+\cos 2 	heta)+3

Vedclass · Accepted Answer

$16$

Vedclass · Answer

$7 \cos ^{2} 	heta-3 \sin ^{2} 	heta-2 \cos ^{2} 2 	heta=2$

$4 \cos ^{2} 	heta+3 \cos 2 	heta-2 \cos ^{2} 2 	heta=2$

$2(1+\cos 2 	heta)+3 \cos 2 	heta-2 \cos ^{2} 2 	heta=2$

$2 \cos ^{2} 2 	heta-5 \cos 2 	heta=0$

$\cos 2 	heta(2 \cos 2 	heta-5)=0$

$\cos 2 	heta=0$

$2 	heta=(2 n+1) \frac{\pi}{2}$

$	heta=(2 n+1) \frac{\pi}{4}$

$S=\left\{\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}ight\}$

For all four values of $	heta$

$X^{2}-2\left(	an ^{2} 	heta+\cot ^{2} 	hetaight) x+6 \sin ^{2} 	heta=0$

$x^{2}-4 x+3=0$

Sum of roots of all four equations $=4 	imes 4=16$.

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$...$

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