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For x in (0,(3)/(2)), " let " f(x)=sqrt(...

For `x in (0,(3)/(2)), " let " f(x)=sqrt(x),g(x) =tan x and h(x)=(1-x^(2))/(1+x^(2))`.
If `phi(x)=((hof)og)(x), " then " phi ((pi)/(3))` is equal to

A

` "tan"(pi)/(12)`

B

` "tan"(11pi)/(12)`

C

` "tan"(7pi)/(12)`

D

` "tan"(5pi)/(12)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given, for `x in (0,3//2)` functions
`f(x)=sqrt(x) " …(i)" `
` g(x)-tanx " …(ii)" `
`and h(x) =(1-x^(2))/(1+x^(2)) " …(iii)" `
Also given `phi(x)=((hof)og)(x)=(hof)(g(x))`
`=h(f(g(x)))`
`=h(f(tanx))`
`=h(sqrt(tanx))=(1-(sqrt(tan x))^(2))/(1+(sqrt(tan x))^(2))`
`=(1-tanx)/(1+tan x)=tan((pi)/(4)-x)`
Now, `phi ((pi)/(3))=tan((pi)/(4)-(pi)/(3))`
`=tan((3pi-4pi)/(12))=tan(-(pi)/(12))`
`= -tan((pi)/(12))=tan(pi-(pi)/(12))`
`=tan((11pi)/(12))`
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