If `y=tan^(2)theta+sectheta, thetane(2n+1)pi//2,then`

Options

A. `y in(-oo,1]`
B. `yin(-oo,-1]`
C. `yin[-1,oo)`
D. none of these

Answer

C

Explanation

We have,
`y=tan^(2)theta+sectheta`
`impliessec^(2)thetasectheta-(y+1)=0`
`=sectheta=(-1pmsqrt(4y+5))/(2)`
Since sec `theta` is real and `sec theta ge1 or, sec thetale-1.`
`therefore 4y+5ge0and((-1+sqrt(4y+5))/(2)geor,(-1-sqrt(4u+5))/(2)le-1)`
`impliesy ge-5/4and (4y+5ge9or,4y+5ge1)`
`implies(yge-5/4and yge1)or,(yge-5/4and yge-1)`
`implies(y ge1)or, (y ge-1)impliesyge-1impliesyin[-1,oo)`