Question

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Answer 1

Answer:

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Answer 2

$</p><p>\mathtt{Question:} </p><p>\\ \\</p><p>\sqrt{\frac{ \cosec A + 1 }{ \cosec A - 1 }} - \sqrt{\frac{ \cosec A - 1 }{ \cosec A + 1}} = 2 \; \tan A </p><p>\\ \\</p><p>\mathtt{Solving \; L.H.S \; separately} </p><p>\\ \\</p><p></p><p>\rightarrow \quad \sqrt{\frac{ \cosec A + 1 }{ \cosec A - 1 }} - \sqrt{\frac{ \cosec A - 1 }{ \cosec A + 1}}</p><p></p><p>\\ \\</p><p>\mathtt{Multiply \; and \; divide \; \sqrt{ \cosec A + 1 } } </p><p>\\ \\</p><p>\rightarrow \quad \sqrt{\frac{ \cosec A + 1 }{ \cosec A - 1 }} \times \sqrt{\frac{ \cosec A + 1 }{ \cosec A + 1 }} - \sqrt{\frac{ \cosec A - 1 }{ \cosec A + 1}} \times \sqrt{\frac{ \cosec A + 1 }{ \cosec A + 1 }}</p><p></p><p>\\ \\</p><p>\rightarrow \quad \sqrt{ \frac{ \cosec A + 1 }{ \cosec A - 1 } \times \frac{ \cosec A + 1 }{ \cosec A + 1 } } - \sqrt{\frac{ \cosec A - 1 }{ \cosec A + 1} \times \frac{ \cosec A + 1 }{ \cosec A + 1 } }</p><p></p><p>\\ \\</p><p>\rightarrow \quad \sqrt{ \frac{ \left( \cosec A + 1 \right)^{2} }{ \cosec^{2} A - 1 } } - \sqrt{ \frac{ \cosec^{2} A - 1 }{ \left( \cosec A + 1 \right)^{2} } } \qquad \left( \because (a+b)(a-b) = a^{2} - b^{2} \right)</p><p></p><p>\\ \\</p><p>\rightarrow \quad \sqrt{ \frac{ \left( \cosec A + 1 \right)^{2} }{ \cancel{1} + \cot^{2} A \cancel{- 1} } } - \sqrt{ \frac{ \cancel{1} + \cot^{2} A \cancel{-1} }{ \left( \cosec A + 1 \right)^{2} } } \qquad \left( \because \; 1 + \cot^{2} A = \cosec^{2} A \right)</p><p>\\ \\</p><p>\rightarrow \quad \sqrt{ \frac{ \left( \cosec A + 1 \right)^{2} }{ \cot^{2} A } } - \sqrt{ \frac{\cot^{2} A }{ \left( \cosec A + 1 \right)^{2} } }</p><p>\\ \\</p><p>\rightarrow \quad \frac{1 + \cosec A}{ \cot A } - \frac{\cot A}{ 1 + \cosec A}</p><p>\\ \\</p><p>\rightarrow \quad \frac{ \left( 1 + \cosec A \right)^{2} - \cot^{2} A }{\cot A \left( 1 + \cosec A \right) }</p><p></p><p>\\ \\</p><p>\rightarrow \quad \frac{ 1 \cancel{+ \cosec^{2} A} + 2 \cosec A \cancel{- \cosec^{2} A} + 1 }{\cot A \left( 1 + \cosec A \right) } \qquad \left( \because 1 + \cot^{2} A = \cosec^{2} A \right)</p><p>\\ \\</p><p>\rightarrow \quad \frac{2 + 2 \cosec A }{ \cot A \left( 1 + \cosec A \right) } </p><p>\\ \\</p><p>\rightarrow \quad \frac{ 2 \cancel{\left( 1 + \cosec A \right)} }{ \cot A \cancel{\left( 1 + \cosec A \right)} }</p><p>\\ \\</p><p>\rightarrow \quad 2 \times \frac{1}{\cot A } </p><p>\\ \\</p><p>\rightarrow \quad 2 \; \tan A \qquad \left( \because \tan A = \frac{1}{\cot A } \right)</p><p>\\ \\ </p><p>\bold{= R.H.S }</p><p>\\ \\</p><p>\bold{Hence, Proved} </p><p>$