Let \(\alpha =3+4+8+9+13+14+\ldots\) upto \(40\) terms. If \((\tan \beta {)}^{\frac{\alpha }{1020}}\) is a root of the e…

Let \(\alpha =3+4+8+9+13+14+\ldots\) upto \(40\) terms. If \((\tan \beta {)}^{\frac{\alpha }{1020}}\) is a root of the equation \({x}^{2}+x-2=0,\beta \in \left(0,\frac{\pi }{2}\right)\), then \({\sin }^{2}\beta +3{\cos }^{2}\beta\) is equal to:

[JEE Main 2026, 8 Apr (Shift 2)]