1.

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

Answer» Correct Answer - A::B
`f(cos 4theta) = (2)/(2-sec^(2) theta)" ...(i)"`
At `cos 4 theta = (1)/(3)`
`implies 2cos^(2)2 theta-1= (1)/(3)`
`impliescos^(2)2 theta=(2)/(3)`
`implies cos2 theta = pmsqrt((2)/(3)) " ...(ii)"`
`therefore f(cos 4 theta) = (2 cos^(2)theta)/(2 cos^(2)theta-1)`
`=(1+cos2 theta)/(cos 2 theta)`
`implies f((1)/(3)) = 1 pm sqrt((3)/(2)) " [from Eq. (ii)]"`


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