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If f(x) = tan^(-1) ((2cot^(2)x)/(1 + cos...

If `f(x) = tan^(-1) ((2cot^(2)x)/(1 + cos^(2)x))` then `d/(dx) (f(f(x)))` at `x = pi/2` is

A

0

B

1

C

`-1`

D

`1/2`

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x) = tan^(-1)((2cot^2 x)/(1+cos^2 x))`
`= tan^(-1)((2 cos^2 x)/(1-cos^4 x))`
`=2 tan^(-1)(cos^(2)x)`
`f'(x) = (-2)/(1+cos^4 x) [ 2 sin x cos x]`
`f'(x) = (4 sin x. Cos x)/(1 + sin^4 x)`
`f' ((pi)/2) = 0`
`d/(dx) f(f(x)) - f'(f(x)) f'(x)`
`d/(dx) f(f(x))` at `(x - pi/2) = f'(f((pi)/2))f'(pi/2) = 0`.
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