1.

Tan A +sec A -1 / tan A -Sec A +1 = 1+ sin A / cos A ​

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EXPLANATION.

\sf : \implies \: \dfrac{ \tan(a) + \sec(a) - 1}{ \tan(a) - \sec(a) + 1} = \dfrac{1 + \sin(a) }{ \cos(a) } \\ \\ \sf : \implies \: 1 = (\sec {}^{2} (a) - \tan {}^{2} (a) ) \\ \\ \sf : \implies \: \frac{ \tan(a) + \sec(a) - ( \sec {}^{2} (a) - \tan {}^{2} (a)) }{ \tan(a) - \sec(a) + 1} \\ \\ \sf : \implies \: \frac{ \tan(a) + \sec(a) - [( \sec(a) - \tan(a))( \sec(a) + \tan(a) )]}{ \tan(a) - \sec(a) + 1}

\sf : \implies \: \dfrac{( \sec(a) + \tan(a)(1 - \sec(a) + \tan(a)) }{ \tan(a) - \sec(a) + 1} \\ \\ \sf : \implies \: \sec(a) + \tan(a) \\ \\ \sf : \implies \: \frac{1}{ \cos(a )} + \frac{ \sin(a) }{ \cos(a) } \\ \\ \sf : \implies \: \frac{1 + \sin(a) }{ \cos(a) } = proved


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