Question

Let $\alpha$ and $\beta$ be the roots of quadratic equation, x²sin∅ - x(sin∅cos∅ + 1) + cos∅ = 0 where ∅ $\epsilon$ (0,45) and $\alpha < \beta$. Find the value of :
$\sf {}^{ \infty } \sum_{{n = 0}} \bigg( { \alpha }^{n} + \dfrac{( - 1) {}^{n} }{ \beta {}^{n} } \bigg)$
​

Answer 1

QUESTION

Let $\alpha$ and $\beta$ be the roots of quadratic equation, x²sin∅ - x(sin∅cos∅ + 1) + cos∅ = 0 where ∅ $\epsilon$ (0,45) and $\alpha < \beta$. Find the value of :

$\sf {}^{ \infty } \sum_{{n = 0}} \bigg( { \alpha }^{n} + \dfrac{( - 1) {}^{n} }{ \beta {}^{n} } \bigg)$

ANSWER

$= \sf \dfrac{1}{1 - \cos∅ } + \dfrac{1}{1 + \dfrac{ 1}{ \sin∅ } }$

SOLUTION

First we need to find the value of $\alpha$ and $\beta$

Given -

• x²sin∅ - x(sin∅cos∅ + 1) + cos∅ = 0

• where ∅ $\epsilon$ (0,45) and $\alpha < \beta$

Let's solve it -

• x²sin∅ - x(sin∅cos∅ + 1) + cos∅ = 0

★We will take xsin∅ and -1 common

• xsin∅ (x - cos∅ ) -1 (x - cos∅) = 0

★ We will take x - cos∅ common

• ( xsin∅-1 ) ( x - cos∅ ) = 0

★ Now ( xsin∅-1 ) = 0 ; ( x - cos∅ ) = 0

∴ $\alpha$ = Cos∅ and $\beta$ = Cosec∅

( ∵ For 0 < θ < 40 ,1 √2 < cosθ < √2 < cosecθ < ∞ ⇒cosθ < cosecθ)

We know that $\alpha < \beta$

Now,

$\sf {}^{ \infty } \sum_{{n = 0}} \bigg( { \alpha }^{n} + \dfrac{( - 1) {}^{n} }{ \beta {}^{n} } \bigg)$

$\sf = (1 + \alpha + { \alpha }^{2} + { \alpha }^{3} + .. \infty ) + (1 - \frac{1}{ \beta } + \frac{1}{ { \beta }^{2} } - \frac{1}{ { \beta }^{3} } + .. \infty )$

$= \sf \dfrac{1}{1 - \alpha } + \dfrac{1}{1 - ( \dfrac{ - 1}{ \beta } )}$

$= \sf \dfrac{1}{1 - \alpha } + \dfrac{1}{1 + \dfrac{ 1}{ \beta } }$

Now we know that $\alpha$ = Cos∅ and $\beta$ = Sin∅

$= \sf \dfrac{1}{1 - \cos∅ } + \dfrac{1}{1 + \dfrac{ 1}{ \sin∅ } }$