If `alpha, beta and gamma` are the three angles with `alpha = 2tan^(-1)(sqrt2-1); beta = 3sin^(-1)(1/sqrt2)+sin^(-1)(-1/2)` and `gamma = cos^(-1)(1/3),` then

Options

A. `alpha glt beta lt y`
B. `alpha lt gamma lt beta`
C. `beta lt gamma lt alpha`
D. `gamma lt beta lt alpha`

Explanation

We have
`alpha =2 tan^(-1) (sqrt(2)-1)=2 tan^(-1)(tan(pi)/(8))=2xx(pi)/(8)=(pi)/(4)=cos^(-21)(1)/sqrt(2)`
`beta =3 sin^(-1)(1)/sqrt(2)+sin^(-1)(1/2)=3xx(pi)/(4)-(pi)/(6)=(7pi)/(12)`
clearly `alpha lt beta`
clarly `cos^(-1)x` is a decreasing funtion on [-1,1] and `1//3lt1//sqrt(2)`
we observe that
`0lty lt (pi)/(2)and(pi)/(2)lt betaltpi rarr gamma lt beta`
Thus we have `alpha lt gamma lt beta`