L.H.S. = \((1 + \cot A - cosec A)(1 + \tan A + \sec A)\)
\(= \left(1 + \frac{\cos A}{\sin A} - \frac 1{\sin A}\right) \left(1 + \frac{\sin A}{\cos A} + \frac 1{\cos A}\right)\)
\(= \left(\frac{\sin A + \cos A - 1}{\sin A}\right) \left(\frac{\cos A + \sin A + 1}{\cos A}\right)\)
\(= \frac{(\sin A + \cos A)^2 - 1^2}{\sin A . \cos A}\)
\(= \frac{\sin^2A + \cos^2A + 2\sin A . \cos A - 1}{\sin A . \cos A}\)
\(= \frac{1 + 2\sin A .\cos A - 1}{\sin A.\cos A}\)
\(= 2\)
= R.H.S.
Hence proved.
