Prove that : $(s i n A + c o s e c A)^{2} + (c o s A + s e c A)^{2} = 7 + t a n^{2} A + c o t^{2} A .$

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Solution

L.H.S. $= (s i n A + c o s e c A)^{2} + (c o s A + s e c A)^{2}$

$= s i n^{2} A + c o s e c^{2} A + 2 s i n A c o s e c A + c o s^{2} A + s e c^{2} A + 2 c o s A s e c A$

$= 1 + 2 + 2 + 1 + t a n^{2} A + 1 + c o t^{2} A$

$= 7 + t a n^{2} A + c o t^{2} A$ = R.H.S.

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