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Answer 1

$= = > \: \: \sqrt{ \frac{ \csc x \: - 1 }{ \csc x \: + 1 } } = \frac{1}{ \sec x + \tan x } \\ \\ \mathtt{ Solving \: L.H.S \: seprately \: }\\ \\ = = > \: \: \sqrt{ \frac{ \csc x \: - 1 }{ \csc x \: + 1 } } \times \sqrt{ \frac{ \csc x \: - 1 }{ \csc x \: - 1 } } \\ \\ = = > \: \: \sqrt{ \frac{ {( \csc \: x - 1) }^{2} }{( \csc \: x + 1)( \csc \: x - 1) } } \\ \\ = = > \: \: \sqrt{ \frac{ {( \csc \: x - 1) }^{2} }{ { \csc}^{2} \: x - 1 } } \\ \\ = = > \: \: \sqrt{ \frac{ {( \csc \: x - 1) }^{2} }{ { \cot }^{2} \: x } } \qquad \quad \: ( \because \: 1 + { \cot }^{2} \: x = { \csc}^{2} \: x \: ) \\ \\ = = > \: \: \frac{ \csc \: x \: - 1}{ \cot \: x} \\ \\ = = > \: \: \frac{ \frac{1}{ \sin \: x } - 1 }{ \frac{ \cos \: x }{ \sin \: x } } \\ \\ = = > \: \: \frac{ \frac{1 - \sin \: x }{ \sin \: x } }{ \frac{ \cos \: x }{ \sin \: x } } \\ \\ = = > \: \: \frac{1 - \sin \: x }{ \sin \: x } \times \frac{ \sin \: x }{ \cos \: x } \\ \\ = = > \: \: \frac{1 - \sin \: x }{ \cos \: x } \\ \\ = = > \: \: \sec \: x - \tan \: x \times \frac{ \sec \: x + \tan \: x }{ \sec \: x + \tan \: x } \\ \\ = = > \: \: \frac{ {( \sec \: x) }^{2} - { (\tan \: x) }^{2} }{ \sec \: x + \tan \: x } \\ \\ = = > \: \: \frac{ { \sec}^{2} \: x - { \tan}^{2} \: x }{ \sec \: x + \tan \: x } \\ \\ = = > \: \: \frac{1}{ \sec \: x + \tan \: x } \qquad \quad \: ( \because \: 1 + { \tan}^{2} \: x = { \sec }^{2} \: x \: ) \\ \\ \mathtt{ \: L.H.S \: = R.H.S} \\ \\ \bold{Hence, Proved}$