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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
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))`
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