The sum of all values of $ heta \in[0,2 \pi]$ satisfying $2 \sin ^2 heta=\cos 2 heta$ and $2 \cos ^2 heta=3 \sin heta$ is

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The sum of all values of $	heta \in[0,2 \pi]$ satisfying $2 \sin ^2 	heta=\cos 2 	heta$ and $2 \cos ^2 	heta=3 \sin 	heta$ is

Aicharya · Accepted Answer

$\begin{aligned} & 2 \sin ^2 	heta=\cos 2 	heta \ & 2 \sin ^2 	heta=1-2 \sin ^2 	heta \ & 4 \sin ^2 	heta=1 \ & \sin ^2 	heta=\frac{1}{4} \ & \sin 	heta= \pm \frac{1}{2} \ & 2 \cos ^2 	heta=3 \sin 	heta \ & 2-2 \sin ^2 	heta+3 \sin 	heta-2=0 \ & (2 \sin 	heta-1)(2 \sin 	heta-2)=0 \ & \sin 	heta=\frac{1}{2} \end{aligned}$ so common equation which satisfy both equations is $\sin 	heta=\frac{1}{2}$ $	heta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad(	heta \in[0,2 \pi])$ Sum $=\pi$ ^

The sum of all values of $\theta \in[0,2 \pi]$ satisfying $2 \sin ^2 \theta=\cos 2 \theta$ and $2 \cos ^2…

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