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 :

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 $ < 45^o), and $\alpha $ < $\beta $. Then $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} $ is equal to :

${1 \over {1 + \cos \theta }} + {1 \over {1 - \sin \theta }}$

${1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$

${1 \over {1 - \cos \theta }} - {1 \over {1 + \sin \theta }}$

${1 \over {1 + \cos \theta }} - {1 \over {1 - \sin \theta }}$

Solution

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