If  $f_n\left(x\right)=\left(\tan \frac{x}{2}\right)(1+\sec x)(1+\sec 2x)\cdots (1+\sec 2^{n}x)$, then among the following is not true

$f_n\left(x\right)=\left(\tan \frac{x}{2}\right)(1+\sec x)(1+\sec 2x)\cdots (1+\sec 2^{n}x)$

$f^n\left(x\right)=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\left(\frac{2{\cos }^2\frac{x}{2}}{\cos x}\right)\left(\frac{2{\cos }^{2}x}{\cos 2x}\right)\cdots \frac{2\cos 2^{n-1}x}{\cos 2^{n}x}$

$=\frac{2^n\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)\cos x\cdots \cos \left(2^{n-1}x\right)}{\cos 2^{n}x}$

$=\tan \left(2^{n}x\right)$

$\therefore f_5\left(\frac{\pi }{128}\right)=\tan (2^5\times \frac{\pi }{128})=\tan \left(\frac{32\pi }{128}\right)=1$