Question

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Answer 1

$Answer \: \\ \\ Given \: \: Question \: \: Is \: \\ \\ \frac{ \sqrt{1 + \cos(x) } }{ \sqrt{1 - \cos(x) } } = ( \csc(x) + \cot(x) ) \\ \\ lhs \: \\ \\ \frac{ \sqrt{1 + \cos(x) } }{ \sqrt{1 - \cos(x) } } \\ \\ Multiply \: Numerator \: \: and \: Denominator \: \: by \: \: \: \\ 1 + \cos(x) \: \: we \: have \\ \\ \frac{ \sqrt{1 + \cos(x) \times (1 + \cos(x)) } }{ \sqrt{1 - \cos(x) \times (1 + \cos(x) ) } } \\ \\ \frac{ \sqrt{(1 + \cos(x)) {}^{2} } }{ \sqrt{1 - \cos {}^{2} (x) } } \\ \\ \frac{1 + \cos(x) }{ \sqrt{ \sin {}^{2} (x) } } \\ \\ \frac{1 + \cos(x) }{ \sin(x) } \\ \\ \frac{1}{ \sin(x) } + \frac{ \cos(x) }{ \sin(x) } \\ \\ \csc(x) + \cot(x) \: \: \: hence \: proved \\ \\ therefore \: \: \\ \\ \frac{ \sqrt{1 + \cos(x) } }{ \sqrt{1 - \cos(x) } } = \csc(x) + \cot(x) \\ \\ Note \: \\ \\ 1 - \cos {}^{2} (x) = \sin {}^{2} (x) \\ \\ \cot(x) = \frac{ \cos(x) }{ \sin(x) } \\ \\ \csc(x) = \frac{1}{ \sin(x) }$