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Mathematics

Algebra

Easy

Question

Let $alpha text and end text beta$ be the roots of the quadratic equation $x squared sin theta minus x left parenthesis sin theta cos theta plus 1 right parenthesis plus cos theta equals 0$ $open parentheses 0 less than theta less than 45 degree close parentheses$ and $alpha less than beta.$ Then, $sum from n equals 0 to infinity of open parentheses alpha to the power of n plus fraction numerator left parenthesis negative 1 right parenthesis to the power of n over denominator beta to the power of n end fraction close parentheses$ is equal to

$\frac{1}{1 - \cos θ} - \frac{1}{1 + \sin θ}$
$\frac{1}{1 - \cos θ} + \frac{1}{1 + \sin θ}$
$\frac{1}{1 + \cos θ} - \frac{1}{1 - \sin θ}$
$\frac{1}{1 + \cos θ} + \frac{1}{1 - \sin θ}$

The correct answer is: $fraction numerator 1 over denominator 1 minus cos theta end fraction plus fraction numerator 1 over denominator 1 plus sin theta end fraction$

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