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

Options

A. `1+-(sqrt3)/(2)`
B. `1+-sqrt((2)/(3))`
C. `1+-sqrt((1)/(3))`
D. `1+-sqrt((1)/(2))`

Answer

A

Explanation

If `0in(0,(pi)/(4))uu((pi)/(4),(pi)/(2)), then 2thetain(0, (pi)/(2))uu((pi)/(2),pi)`
Therefore, `cos 2 theta` can be positive or negative.
Hence, `cos2theta=pmsqrt((1+cos4theta)/(2))`
`impliescos 2theta=pmsqrt((1+(1)/(3))/(2))=+-sqrt((2)/(3))" "["Taking"cos4theta=1/3]`
Now, `f(cos4theta)=(2)/(2-sec^(2)theta)=(2cos^(2)theta)/(2cos^(2)theta-1)=(1+cos2theta)/(cos2theta)`
`impliesf((1)/(3))=1+-sqrt((3)/(2)). [because cos4theta=1/3and cos2theta=+-sqrt((2)/(3))]`