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Let $\alpha$ and $\beta$ be the roots of the quadratic equation

$x^{2}\sin \theta -x (\sin \theta \cos \theta + 1)+\cos \theta =0\: \: (0< \theta < 45^{\circ}),\: and\: \alpha < \beta .$

Then

$\sum_{n=0}^{\infty} (\alpha ^{n}+\frac{(-1)^{n}}{\beta ^{n}})$ is equal to :
Option 1)
$\frac{1}{1+\cos \theta }+\frac{1}{1-\sin \theta }$
Option 2)
$\frac{1}{1-\cos \theta }-\frac{1}{1+\sin \theta }$
Option 3)
$\frac{1}{1-\cos \theta }+\frac{1}{1+\sin \theta }$
Option 4)
$\frac{1}{1+\cos \theta }-\frac{1}{1-\sin \theta }$

Answers (1)

best_answer

Roots of Quadratic Equation -

$\alpha = \frac{-b+\sqrt{b^{2}-4ac}}{2a}$

$\beta = \frac{-b-\sqrt{b^{2}-4ac}}{2a}$

- wherein

$ax^{2}+bx+c= 0$

is the equation

$a,b,c\in R,\: \: a\neq 0$

9

Roots of quadratic equation,

$ax^{2}+bx+c=0$

$\alpha ,\beta =\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

$a=\sin \theta$

$b=-(1+\sin \theta \cos \theta )$

$c=\cos \theta$

$\alpha ,\beta =\frac{(1+\sin \theta \cos \theta )\pm \sqrt{(1+\sin \theta \cos \theta)^{2}-4\sin \theta \cos \theta }}{2\sin \theta }$

$\alpha =\cos \theta \: \: ,\: \: \beta =\csc \theta$

$\sum_{n=0}^{\infty }\left [ \alpha ^{n} + \frac{(-1)^{n}}{\beta ^{n}}\right ]=\sum_{n=0}^{\infty }\left [ (\cos \theta) ^{n} +(-sin\theta )^{n}\right ]$

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

Option 1)

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

Option 2)

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

Option 3)
$\frac{1}{1-\cos \theta }+\frac{1}{1+\sin \theta }$
Option 4)

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

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