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`sum_(n=0)^(oo) (alpha^(n)+((-1)^(n))/(beta^(n)))` is to equal toA. `(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)`

Let `alpha` and `beta` be the roots of the quadratic equation `x sin^(

1.	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`sum_(n=0)^(oo) (alpha^(n)+((-1)^(n))/(beta^(n)))` is to equal toA. `(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» Correct Answer - B 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}`