If the value of \[ \sum _{ r=1 }^{ 100 }{ \frac { \tan\quad { 2 }^{ r-1 } }{ \cos\quad { 2 }^{ r } } } =\quad \tan\quad { 4 }^{ n }\quad -\quad \tan\quad 1\] , find n, where 'n' is a natural number.

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