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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)

best_answer

$f(x)=\sqrt{x}$

$g(x)=\tan x$

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

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

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

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

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

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

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

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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