Trig1.7

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Prove that \frac{\csc A}{\csc A-1}+\frac{\csc A}{\csc A+1}=2\sec^2 A\,

LHS= \frac{\csc A}{\csc A-1}+\frac{\csc A}{\csc A+1}=\frac{\csc A (\csc A+1)+\csc A (\csc A-1)}{(\csc A+1)(\csc A-1)}=\frac{\csc^2 A+\csc^2 A+\csc A-\csc A}{\csc^2 A-1}\,

\frac{2\csc^2 A}{\cot^2 A}=\frac{2\csc^2 A}{\frac{\cos^2 A}{\sin^2 A}}\,

2\frac{\sin^2 A}{(\cos^2 A)(\sin^2 A)}=2\frac{1}{\cos^2 A}=2\sec^2 A\,=RHS


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