So much inverse

Calculus Level pending

\(f\left( x \right) =\arctan { \left[ \frac { \sqrt { 1+\sin { x } } +\sqrt { 1-\sin { x } } }{ \sqrt { 1+\sin { x } } -\sqrt { 1-\sin { x } } } \right] } \\ If\quad f^{ ' }\left( \frac { \pi }{ 4 } \right) =\frac { A }{ B } \\ Find\quad A+B\)

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