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
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
- [JEE MAIN 2019]
- 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
$\frac{1}{{1 + \cos \,\theta }} - \frac{1}{{1 - \sin \,\theta \,}}$
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