The value of `(alpha^3)/2cos e c^2(1/2tan^(-1)alpha/beta)+(beta^3)/2sec^2(1/2tan^(-1)(beta/alpha))i se q u a lto` `(alpha+beta)(alpha^2+beta^2)` (b) `(alpha+beta)(alpha^2-beta^2)` `(alpha+beta)(alpha^2+beta^2)` (d) none of these

Options

A. `(alpha - beta) (alpha^(2) + beta^(2))`
B. `(alpha + beta) (alpha^(2) - beta^(2))`
C. `(alpha + beta) (alpha^(2) + beta^(2))`
D. none of these

Answer

C

Explanation

`(alpha^(3))/(2) cosec^(2) ((1)/(2) tan^(-1). (alpha)/(beta)) + (beta^(3))/(2) sec^(2) ((1)/(2) tan^(-1).(beta)/(alpha))`
`= alpha^(3) (1)/(1 - cos(tan^(-1) ((alpha)/(beta)))) + beta^(3) (1)/(1 + cos (tan^(-1).(beta)/(alpha)))`
`= alpha^(3) (1)/(1 -cos (cos^(-1) ((beta)/(sqrt(alpha^(2) + beta^(2)))))) + beta^(3) (1)/(1 + cos (cos^(-1).(alpha)/(sqrt(alpha^(2) + beta^(2)))))`
`= alpha^(3) (1)/(1 - (beta)/(sqrt(alpha^(2) + beta^(2)))) + beta^(3) (1)/(1 + (alpha)/(sqrt(alpha^(2) + beta^(2))))`
`= sqrt(alpha^(2) + beta^(2)) ((alpha^(3))/(sqrt(alpha^(2) + beta^(2)) - beta) + (beta^(3))/(sqrt(alpha^(2) + beta^(2)) + alpha))`
`= sqrt(alpha^(2) + beta^(2)) (alpha^(3) (sqrt(alpha^(2) + beta^(2)) + beta)/(alpha^(2)) + beta^(3) (sqrt(alpha^(2) + beta^(2)) - alpha)/(beta^(2)))`
`= sqrt(alpha^(2) + beta^(2)) [alpha(sqrt(alpha^(2) + beta^(2)) + beta) + beta (sqrt(alpha^(2) + beta^(2)) - alpha)]`
`= sqrt(alpha^(2) + beta^(2)) (alpha + beta) sqrt(alpha^(2) + beta^(2))`
`= (alpha + beta) (alpha^(2) + beta^(2))`