Evaluate the following : (1 + tan A + sec A) (1 + cot A - cosec A).

The given expression is

[latex](1+\tan A+\sec A)(1+\cot A-{cosec} A) \\ =\left(1+\frac{\sin A}{\cos A}+\frac{1}{\cos A}\right)\left(1+\frac{\cos A}{\sin A}-\frac{1}{\sin A}\right) \\ =\left(\frac{\cos A+\sin A+1}{\cos A}\right)\left(\frac{\sin A+\cos A-1}{\sin A}\right) \\ =\frac{(\cos A+\sin A+1)(\sin A+\cos A-1)}{\sin A \cos A} \\ =\frac{(\cos A+\sin A)^{2}-1^{2}}{\sin A \cos A} \\ =\frac{\cos ^{2} A+\sin ^{2} A+2 \sin A \cos A-1}{\sin A \cos A} \\ =\frac{1+2 \sin A \cos A-1}{\sin A \cos A} \\ =\frac{2 \sin A \cos A}{\sin A \cos A} \\ =2[/latex]

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