Trigonometric summation

Geometry Level 5

\[\displaystyle{f^{ n }\left( \theta \right) =\sum _{ k=0 }^{ n }{ { 4 }^{ k }\sec ^{ 2 }{ \left( { 2 }^{ k }\theta \right) } } }\] Define a function \(f\) as above and if \[\displaystyle{f^{ 504 }\left( \cfrac { \pi }{ { 2 }^{ 507 } } \right) \equiv { 2 }^{ a }-\csc ^{ 2 }{ \left( \cfrac { \pi }{ { 2 }^{ b } } \right) } }\] for positive integers \(a,b\) . Evaluate \(a-b\).

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