If `y=(sin^4x-cos^4x+sin^2xcos^2x)/(sin^4x+cos^4x+sin^2xcos^2x),x in (0,pi/2)` , then `-3/2lt=ylt=1/2` (b) `1lt=ylt=1/2` `-5/3lt=y1` (d) none of these

Options

A. `-(3)/(2)le y le (1)/(2)`
B. `(1)/(2)le y le 1`
C. `-(5)/(3)le y le 1`
D. `-1le y le 1`

Answer

D

Explanation

`y = (sin^(4)x-cos^(4)x+sin^(2)x cos^(2)x)/(sin^(4)x +cos^(4)x +sin^(2)x cos^(2)x cos^(2)x)`
`y = (- cos2x+(1)/(4)sin^(2)2x)/(1-(1)/(4)sin^(2)2x)`
`= (1-4cos2x- cos^(2)2x)/(3+cos^(2)2x)`
`(1+y)cos^(2)2x+4cos2x+3y-1=0`
`D ge 0`
`-(5)/(3) le y le 1`
but when `y = 1`
`x = (pi)/(2)` which is not permissible