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LHS = \frac{\sec A + \tan A - 1}{1 - \sec A + \tan A} = \frac{(\sec A + \tan A) - 1}{(1 - \sec A + \tan A)} = \frac{(\sec A + \tan A) - (\sec^{2} A - \tan^{2} A)}{(1 - \sec A + \tan A)} = \frac{(\sec A + \tan A) - (\sec A - \tan A) (\sec A + \tan A)}{(1 - \sec A + \tan A)} = \frac{(\sec A + \tan A) (1 - \sec A + \tan A)}{(1 - \sec A + \tan A)} = \sec A + \tan A = \frac{1}{\cos A} + \frac{\sin A}{\cos A} = \frac{1 + \sin A}{\cos A} = \frac{1}{\cos A} + \frac{\sin A}{\cos A} = \sec A + \tan A = RHS

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