If `|2 sin theta-cosec theta| ge 1` and `theta ne (n pi)/2, n in Z`, then

Options

A. `cos 2 theta ge 1//2`
B. `cos 2 theta ge 1//4`
C. `cos 2 theta le 1//2`
D. `cos 2 theta le 1//4`

Answer

A

Explanation

`|2 sin theta- cosec theta| ge 1`
or `|2 sin^(2) theta-1| ge |sin theta|`
or `|cos 2 theta| ge | sin theta|`
or `2 cos^(2) 2 theta ge 1-cos 2 theta`
or `2 cos^(2) 2 theta+ cos 2 theta-1 ge 0`
or `(2 cos 2 theta-1) (cos 2 theta+1) ge 0`
`rArr cos 2 theta ge 1/2" "[" as" cos theta ne 0, i.e., cos 2 theta ne -1]`