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Let f:(-1,1)vecR be such that f(cos4thet...

Let `f:(-1,1)vecR` 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) `1-sqrt(3/2)` (b) `1+sqrt(3/2)` (c) `1-sqrt(2/3)` (d) `1+sqrt(2/3)`

A

`1-sqrt((3)/(2))`

B

`1+sqrt((3)/(2))`

C

`1-sqrt((2)/(3))`

D

`1+sqrt((2)/(3))`

Text Solution

Verified by Experts

The correct Answer is:
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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