Explanation
$$ \left( \dfrac { 1-\sin \alpha }{ \cos \alpha } +\dfrac { \cos \alpha }{ 1+\sin \alpha } \right) \left( \sec \alpha \quad +\quad \dfrac { 1 }{ \cot \alpha } \right) \\= \left( \dfrac { (1-\sin \alpha )(1+\sin \alpha )+\cos ^ 2 \alpha \quad }{ (1+\sin \alpha )\cos \alpha } \right) \left( \dfrac { 1 }{ \cos \alpha } +\quad \dfrac { \sin \alpha }{ \cos \alpha } \right) $$ $$\\ = \left( \dfrac { (1-\sin ^{ 2 }\alpha +\cos ^{ 2 }\alpha \quad }{ (1+\sin \alpha )\cos \alpha } \right) \left( \dfrac { 1+\sin \alpha }{ \cos \alpha } \right) \\= \left( \dfrac { \cos ^{ 2 }\alpha +\cos ^{ 2 }\alpha \quad }{ \cos \alpha } \right) \left( \dfrac { 1 }{ \cos \alpha } \right) \\= 2 $$
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