JEE Main 2026 · 5 Apr · Shift 1, Mathematics question 6

Roots. $\tan A+\tan B=2$, $\tan A\tan B=-5$, so $\tan(A+B)=\dfrac{2}{1+5}=\dfrac13$.

Quadrant. The roots have opposite signs, so $A+B\in(-\tfrac\pi2,\tfrac\pi2)$; with a positive tangent, $A+B\in(0,\tfrac\pi2)$ and $\cos(A+B)=\dfrac3{\sqrt{10}}$.

Half angle.
$$20\sin^2\frac{A+B}2=10\left(1-\frac3{\sqrt{10}}\right)=10-3\sqrt{10}.$$

(C)