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  4. If sin 2 θ and cos 2 θ are solutions of x2+b x-c=0, then

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Q. If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^{2}+b x-c=0$, then

TS EAMCET 2019

Solution:

According to given informations
$\sin \,2 \,\theta+\cos \,2 \,\theta=-b$ and
$\sin \,2 \,\theta \cos\, 2 \,\theta=-c$
$\because (\sin \,2 \,\theta+\cos \,2 \,\theta)^{2}=\sin ^{2} 2 \theta+\cos ^{2} 2 \,\theta+2 \,\sin \,2 \,\theta \,\cos \,2 \,\theta$
$\Rightarrow (-b)^{2}=1+2(-c)$
$\Rightarrow b^{2}+2 c-1=0$