Question

prove that:

√1+cosA/1-cosA=cosecA+cotA​

Answer 1

To prove :

$\star \: \sqrt{ \frac{1 + \cos \: A}{1 - \cos \: A} } = \cosec \: A + \cot \: A$

Take LHS :

$\implies \: \sqrt{ \frac{1 + \cos \: A}{1 - \cos \:A } }$

Rationalise the denominator :

$\implies \: \sqrt{ \frac{1 + \cos \:A }{1 - \cos \: A} \times \frac{1 + \cos \: A}{1 + \cos \:A } } \\ \\ \implies \: \sqrt{ \frac{ {(1 + \cos \: A})^{2} }{ {1}^{2} - { \cos}^{2} A} } \\ \\ \implies \: \sqrt{ \frac{ {(1 + \cos \: A)}^{2} }{ { \sin}^{2}A } } \\ \\ \implies \: \frac{1 + \cos \:A }{ \sin \:A } \\ \\ \implies \: \frac{1}{ \sin \: A} + \frac{ \cos \: A}{ \sin \: A} \\ \\ \implies \: \cosec \: A \: + \cot \: A$

LHS = RHS .

Hence, Proved .

•Formula used :

$\star \boxed{ \bold{ { \sin }^{2} \alpha + { \cos }^{2} \alpha = 1}} \\ \\ \star \boxed{ \bold{ \sin \: \alpha = \frac{1}{ \csc( \alpha ) } }} \\ \\ \star \boxed{ \bold{ \cot \: \alpha = \frac{ \cos( \alpha ) }{ \sin( \alpha ) } }}$

Answer 2

Step-by-step explanation:

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