A 'Too Complicated' Trigo Sum

Algebra Level 4

The sum\[ \sum _{ x=2 }^{ 44 }{ 2.sinx.sin1.\{ 1+\ sec(x-1).\ sec(x+1)\} } \] can be written in the form \[\sum _{ n=1 }^{ 4 }{ { (-1) }^{ n } } .\frac { { \phi }^{ 2 }({ \theta }_{ n }) }{ \Psi ({ \theta }_{ n }) }\] where \[\phi \quad and \quad \psi\] are trigonometric functions and \[{ \theta }_{ 1 }\quad ,\quad { \theta }_{ 2 }\quad ,\quad { \theta }_{ 3 }\quad and\quad { \theta }_{ 4 }\] are in degrees \[\epsilon \quad [0,45]\]. Find \[\frac { { \theta }_{ 1 }+{ \theta }_{ 2 }+{ \theta }_{ 3 }+{ \theta }_{ 4 } }{ 46 } \].

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