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The algebraic expression for f(x)=tan(si...

The algebraic expression for `f(x)=tan(sin^(-1)(cos("tan"^(-1)(x)/(2))))` is

A

`(2)/(x)`

B

`(x)/(2)`

C

`(1)/(x)`

D

`(2)/(|x|)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `"tan"^(-1) (x)/(2)=theta`
`rArr tan. theta=(x)/(2)`
`rArr cos("tan"^(-1)(x)/(2))=cos. theta=(2)/(sqrt(4+x^(2)))`
`rArr f(x)=tan["sin"^(-1)(2)/(sqrt(4+x^(2)))]=(2)/(x)`
If `x gt 0`, then `f(x)=tan("tan"^(-1)(2)/(x))=(2)/(x)`
If `x lt 0`, then `f(x)=tan(tan^(-1)((-2)/(x)))=(-2)/(x)`
`rArr f(x)=(2)/(|x|)`
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