For a positive integer $$\,n$$, let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$$ Then
A
$${f_2}\left( {{\pi \over {16}}} \right) = 1$$
B
$${f_3}\left( {{\pi \over {32}}} \right) = 1$$
C
$${f_4}\left( {{\pi \over {64}}} \right) = 1$$
D
$${f_5}\left( {{\pi \over {128}}} \right) = 1$$