JEE Main 2025 — 22 January, Evening Shift — Question 18

$2 \sin ^{2} \theta=\cos 2 \theta$

$2 \sin ^{2} \theta=1-2 \sin ^{2} \theta$

$4 \sin ^{2} \theta=1$

$\sin ^{2} \theta=\frac{1}{4}$

$\sin \theta= \pm \frac{1}{2}$

$2 \cos ^{2} \theta=3 \sin \theta$

$2-2 \sin ^{2} \theta+3 \sin \theta-2=0$

$(2 \sin \theta-1)(2 \sin \theta-2)=0$

$\sin \theta=\frac{1}{2}$

so common equation which satisfy both equations is

$\sin \theta=\frac{1}{2}$

$\theta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad(\theta \in[0,2 \pi])$

Sum $=\pi$