If `alpha=tan^(-1)((4x-4x^3)/(1-6x^2+x^2)),beta=2sin^(-1)((2x)/(1+x^2))` and `tanpi/8=k ,` then `alpha+beta=piforx in [(1,1)/k]` `alpha+betaforx in (-k , k)` `alpha+beta=piforx in [(1,1)/k]` `alpha+beta=0forx in [-k , k]`

Options

A. `alpha + beta = pi " for " x in [1, (1)/(k))`
B. `alpha = beta " for " x in (-k, k)`
C. `alpha + beta = - pi " for " x in [1, (1)/(k))`
D. `alpha + beta = 0 " for " x in (-k, k)`

Answer

A, B

Explanation

Put `x = tan theta`
`rArr alpha = tan^(-1) (tan 4 theta)`
`= 4 theta - pi " for " x in [1, (1)/(k))`
`= 4 theta " for " x in (-k, k)`
Also, `beta - 2(pi - 2 theta) " for " x in [1, (1)/(k))`
`= 4 theta " for " x in (-k, k)`