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 lt theta lt 45^(@))`, and `alpha lt beta`.
Then `underset(n=0)overset(oo)(Sigma) (alpha^(n) + ((-1)^(n))/(beta^(n)))` is 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

B

Explanation

Given
`x^(2) sin theta -x sin theta cos theta -x + cos theta = 0`
Where `0 lt theta lt 45^(@)`
`sin theta (x- cos theta) - 1( x - costheta) = 0`
`rArr (x- cos theta)(x sin theta-1) = 0`
`rArr x = cos theta, x cosec theta`
`x = cos theta, x = coses theta`
`rArr alpha cos theta and beta = cosec theta`
`(because "For" 0 lt theta lt 40^(@), (1)/(sqrt(2)) lt cos theta lt sqrt(2) lt "cosec " theta lt oo rArr cos theta lt cosec theta)`
Now, consider, `underset(n = 0)overset(oo)sum(alpha^(n)+ ((-1)^(n))/(beta^(n))) = underset(n = 0 )overset(oo)sum + underset(n=0) overset(oo)sum ((-1)^(n))/(beta^(n))`
`= (1 + alpha + alpha^(2) + alpha^(3)+........oo)`
`+ (1-(1)/(beta) +(1)/(beta^(2)) - (1)/(beta^(3)) +.....oo)`
`= (1)/(1-alpha) + (1)/(1-((-1)/(beta)))= (1)/(1-alpha) +(1)/(1+(1)/(beta))`
`= (1)/(1-cos theta) +(1)/(1+ sin theta) " "{because (1)/(beta)sin^(2) = 1-cos^(2)x}`