Let `alpha` and `beta` be the roots of the quadratic equation `x sin^(2) theta -x (sin theta cos theta + 1) + cos theta =0 (0le thetale 45^(@)) and alpha le beta` . Then`underset(n =0)overset(oo)sum(alpha^(n)+((-1)^(n))/(beta^(n)))` is to equal to

Options

A. `(1)/(1-cos theta) + ( 1)/(1- sin theta)`
B. `(1)/( 1 + cos theta) - (1)/( 1 - sin theta)`
C. `(1)/(1- cos theta)+ (1)/(1+ sin theta)`
D. `(1)/(1- cos theta) - ( 1)/( 1 + sin theta)`

Answer

C

Explanation

`because alpha, beta` be the roots of the equation: `" " x ^(2) sin theta - x sin theta cos theta - x + cos theta = 0`
`implies x sin theta ( x- cos theta) -1 (x -cos theta)=0 implies " "( x sin theta -1) (x - cos theta) =0implies x = (1)/(sin theta), cos theta`
Given that `0 lt theta lt 45^(@) and alpha lt beta" " therefore " " alpha =cos theta beta = (1)/(sin theta)`
Now, `sum _(n =0) ^(oo) (alpha ^(n) + ((-1)^(n))/(beta ^(n))) = sum _(n =0) ^(oo) alpha ^(n) + sum_(n =0) ^(oo) ((-1)^(n))/(beta ^(n))= (1)/(1 - alpha ) + (beta)/(1 + beta) = (1)/(1 - cos theta) +((1)/(sin theta))/(1 + (1)/(sin theta ))= (1)/(1-cos theta) + (1)/(1 +sin theta)`