Let `f : (-1, 1) -> R` be such that `f(cos4theta) = 2/(2-sec^2theta` for `theta in (0, pi/4) uu (pi/4, pi/2)`. Then the value(s) of `f(1/3)` is/are

Options

A. `1-sqrt((3)/(2))`
B. `1+sqrt((3)/(2))`
C. `1-sqrt((2)/(3))`
D. `1+sqrt((2)/(3))`

Answer

A, B

Explanation

For `theta in ( 0, (pi)/(4)) uu ((pi)/(4), (pi)/(2))`
Let` " " cos 4 theta = (1)/(3)`
`rArr cos 2 theta = pm sqrt((1+ cos 4 theta)/(2)) = pm sqrt((2)/(3))`
`f((1)/(3)) = (2)/(2-sec^(2) theta) = ( 2cos^(2) theta)/( 2cos ^(2) theta -1) = 1 + (1)/( cos 2 theta)`
`rArr f((1)/(3)) = 1 - sqrt((3)/(2)) or 1+ sqrt((3)/(2))`