If `alpha,beta,gamma` are ccute angles and `costheta=sinbeta//sinalpha,cosvarphi=singammasinalphaa n dcos(theta-varphi)=sinbetasingamma` , then the value of `tan^2alpha-tan^2beta-tan^2gamma` is equal to `-1` (b) `0` (c) `1` (d) 2

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

B

Explanation

Let `y=cosx[sinx+sqrt(sin^(2)x+sin^(2)((pi)/(6))]` is
`rArr y^(2)tan^(2)x-2ytan x+(y^(2)-(1)/(4))ge0`
For real `tan x`,
disc `ge0`
`4y^(2)-4y^(2)(y^(2)-(1)/(4))ge0`
`rArr 0ley^(2)le(5)/(4)`
`therefore` Maximum vlaue of `y=sqrt(5)/(2)`.