Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation \( x^2 \sin \theta - x (\sin \theta \cos \theta + 1) + \cos \theta = 0 \)
\( (0 < \theta < 45^\circ) \)
and \( \alpha < \beta \). Then \( \sum_{n=0}^{\infty} \left( a^n + \dfrac{(-1)^n}{\beta^n} \right) \) is equal to:1
\( \dfrac{1}{1 - \cos \theta} + \dfrac{1}{1 + \sin \theta} \)
2
\( \dfrac{1}{1 + \cos \theta} + \dfrac{1}{1 - \sin \theta} \)
3
\( \dfrac{1}{1 - \cos \theta} - \dfrac{1}{1 + \sin \theta} \)
4
\( \dfrac{1}{1 + \cos \theta} - \dfrac{1}{1 - \sin \theta} \)