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

• A

  $\frac{1}{{1 - \cos \,\theta }} - \frac{1}{{1 + \sin \,\theta \,}}$

• B

  $\frac{1}{{1 + \cos \,\theta }} + \frac{1}{{1 - \sin \,\theta \,}}$

• C

  $\frac{1}{{1 - \cos \,\theta }} + \frac{1}{{1 + \sin \,\theta \,}}$

• D

  $\frac{1}{{1 + \cos \,\theta }} - \frac{1}{{1 - \sin \,\theta \,}}$