JEE Main 2025 (Online) 22nd January Evening ShiftQuestion 1
The sum of all values of $\theta\in[0,2\pi]$ satisfying $2\sin^{2}\theta=\cos2\theta$ and $2\cos^{2}\theta=3\sin\theta$ is
Trigonometric Ratio & Identities
Solution
Concept. Write both equations in $\sin\theta$ and keep the common solutions. Formulas used. (1) $\cos2\theta=1-2\sin^2\theta$ and $\cos^2\theta=1-\sin^2\theta$Step 1 - First equation. $2\sin^2\theta=1-2\sin^2\theta$, so $\sin^2\theta=\frac14$. Step 2 - Second equation. $2-2\sin^2\theta=3\sin\theta$, i.e. $(2\sin\theta-1)(\sin\theta+2)=0$, so $\sin\theta=\frac12$. Step 3 - Common values. $\sin\theta=\frac12$ satisfies both: $\theta=\frac\pi6,\frac{5\pi}6$. Step 4 - Sum. $\pi$. Answer: (A)