There exists a positive real number of x...

There exists a positive real number of `x` satisfying `"cos"(tan^(-1)x)=xdot` Then the value of `cos^(-1)((x^2)/2)i s` `pi/(10)` (b) `pi/5` (c) `(2pi)/5` (d) `(4pi)/5`

A

`(pi)/(10)`

B

`(pi)/(5)`

C

`(2pi)/(5)`

D

`(4pi)/(5)`

Text Solution

Verified by Experts

The correct Answer is:

C

Let `tan^(-1) (x) = theta " or " x = tan theta`
`rArr cos theta = x " " rArr (1)/(sqrt(1 + x^(2))) = x` ltrbgt `rArr x^(2) (1 + x^(2)) = 1 rArr x^(2) = (-1 +- sqrt5)/(2)`
So, `x^(2) = (sqrt5 -1)/(2) rArr (x^(2))/(2) = (sqrt5 -1)/(4)`
Now `cos^(-1) ((sqrt5 -1)/(4)) = cos^(-1) (sin.(pi)/(10)) = cos^(-1) (cos.(2pi)/(5)) = (2pi)/(5)`

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