For x\equiv \left ( 0,\frac{3}{2} \right ), let f(x)=\sqrt{x},g(x)=\tan x and h(x)=\frac{1-x^{2}}{1+x^{2}}. If \phi \left ( x \right )=\left ( \left (hof \right )og\left ( x \right ) \right ), then \phi \left ( \frac{\pi }{3} \right ) is equal to : 

 

  • Option 1)

    \tan \frac{11\pi }{12}   

  • Option 2)

       \tan \frac{7\pi }{12}         

  • Option 3)

     \tan \frac{5\pi }{12}     

  • Option 4)

     \tan \frac{\pi }{12}

 

Answers (1)

f(x)=\sqrt{x}

g(x)=\tan x

h(x)=\frac{1-x^{2}}{1+x^{2}}

fog(x)=\sqrt{\tan x}

hofog(x)=h(\sqrt{\tan x})=\frac{1-\tan x}{1+\tan x}=\tan \left ( \frac{\pi }{4}-x \right )

\phi \left ( x \right )=\tan \left ( \frac{\pi }{4}-x \right )

\phi \left ( \frac{\pi }{3} \right )=\tan \left ( \frac{\pi }{4}-\frac{\pi }{3} \right )=\tan \left ( -\frac{\pi }{12} \right )=-\tan \frac{\pi }{12}

               =\tan \left ( \pi -\frac{\pi }{12} \right )

               =\tan \frac{11\pi }{12}


Option 1)

\tan \frac{11\pi }{12}   

Option 2)

   \tan \frac{7\pi }{12}         

Option 3)

 \tan \frac{5\pi }{12}     

Option 4)

 \tan \frac{\pi }{12}

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