Question

Prove that: (secA-cosec A) (1+cot A +tan A) =( sec^2A/cosecA)-(Cosec^2A/secA)
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Answer 1

Explanation:

`(\sec A - \csc A)(1 + \cot A + \tan A)`

`=(\sec A - \csc A)\left(1 + (\cos A)/(\sin A) + (\sin A)/(\cos A) \right)`

`=(\sec A - \csc A)\left(1 + (\cos^2 A + \sin^2 A)/(\sin A\cos A) \right)`

`=(\sec A - \csc A)\left((1 + \sin A \cos A)/(\sin A \cos A) \right)`

`=\left(((1)/(\cos A) - (1)/(\sin A)+\sin A - \cos A)/(\sin A\cos A)\right)`

`=(\sin A - \sin A \cos^2A - \cos A + \cos A\sin^2A)/((\sin A\cos A)^2)`

`=(\sin A(1 - \cos^2A) - \cos A (1 - \sin^2 A))/((\sin A\cos A)^2)`

`=(\sin^3A - \cos^3A)/(\sin^2A\cos^2A)`

`=(\sin A)/(\cos^2A) - (\cos A)/(\sin^2A)`

`=\left((1)/(\cos A)\right)\left((\sin A)/(1)\right) - \left((1)/(\sin^2A)\right) \left((\cos A)/(1)\right)`

`=\sec^2A\csc A - \csc^2A\sec A`