Sequences & Series · Sum of arithmetic progressions

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, \dfrac{\pi}{2}\right)

, then

\sin^2\beta + 3\cos^2\beta

is equal to:

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