JEE Math Practice Question

Let $$\alpha $$ and $$\beta $$ be the roots of the quadratic equation x2 sin $$\theta $$ – x(sin $$\theta $$ cos $$\theta $$ + 1) + cos $$\theta $$ = 0 (0 < $$\theta $$ < 45o), and $$\alpha $$ < $$\beta $$. Then $$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} $$ is equal to :

1. A.$${1 \over {1 + \cos \theta }} + {1 \over {1 - \sin \theta }}$$
2. B.$${1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$
3. C.$${1 \over {1 - \cos \theta }} - {1 \over {1 + \sin \theta }}$$
4. D.$${1 \over {1 + \cos \theta }} - {1 \over {1 - \sin \theta }}$$