Let $\alpha$ and $\beta$ the roots of the quadratic equation $x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)$, and $\alpha<\beta$.
Then $\sum_{n=0}^{\infty}\left(\alpha^n+\frac{(-1)^n}{\beta^n}\right)$ is equal to :
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