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જો $\alpha $ અને $\beta $ એ દ્રીઘાત સમીકરણ ${x^2}\,\sin \,\theta - x\,\left( {\sin \,\theta \cos \,\,\theta + 1} \right) + \cos \,\theta = 0\,\left( {0 < \theta < {{45}^o}} \right)$ ના ઉકેલો હોય અને $\alpha < \beta $ તો $\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + \frac{{{{\left( { - 1} \right)}^n}}}{{{\beta ^n}}}} \right)} $ = ......
$\frac{1}{{1 - \cos \,\theta }} - \frac{1}{{1 + \sin \,\theta \,}}$
$\frac{1}{{1 + \cos \,\theta }} + \frac{1}{{1 - \sin \,\theta \,}}$
$\frac{1}{{1 - \cos \,\theta }} + \frac{1}{{1 + \sin \,\theta \,}}$
$\frac{1}{{1 + \cos \,\theta }} - \frac{1}{{1 - \sin \,\theta \,}}$
Solution
Using quadratic formula,
$x=\frac{(\cos \theta \sin \theta+1) \pm \sqrt{(\cos \theta \sin \theta+1)^{2}-4 \sin \theta \cos \theta}}{2 \sin \theta}$
$=\frac{(\cos \theta \sin \theta+1)^{2} \pm(\cos \theta \sin \theta-1)}{2 \sin \theta}$
$ = \cos \,\theta ,\,\cos ec\,\theta $
$\alpha = \cos \,\theta ,\,\beta = \cos ec\,\theta $
$\therefore \,\sum\limits_{n = 0}^\infty {{\alpha ^n}} + \frac{{{{( – 1)}^n}}}{{{\beta ^n}}}$
$ = \sum\limits_{n = 0}^\infty {{{(\cos ec)}^n}} + \sum\limits_{n = 0}^\infty {{{( – \sin \theta )}^n}} $
$=\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}$
$\therefore $ $(C)$ is the correct