Question

Prove the following...​

Answer 1

`{ \rm{ \sqrt{ (1 + \cos(x) )/(1 - \cos(x) ) } }} \\ \\`

- Rationalize the denominator of the above expression;

`{ \rm{ = \sqrt{ ((1 + \cos(x)).(1 + \cos(x)) )/((1 - \cos(x)).(1 + \cos(x)) ) } }} \\ \\ = { \rm \sqrt{ \frac{( {1}^(2) + 2 \cos(x) + \cos ^(2)(x)) }{( {1}^(2) - { \cos }^(2)(x)) } } } \\ \\ = { \rm\sqrt{ \frac{(1 + 2 \cos(x) + { \cos }^(2) (x)) }{(1 - { \cos }^(2)(x)) } } }`

- From the above expression, 1 - cos²x = sin²x

`= { \rm{ \sqrt{ \frac{1 + 2 \cos(x) + { \cos }^(2)(x) }{ { \sin }^(2)(x) } } }} \\`

`= { \rm{ \sqrt{ \frac{ {(1 + \cos(x)) }^(2) }{ { \sin}^(2)x } } }} \\ \\ = { \rm{ \frac{ \sqrt{(1 + \cos(x)) {}^(2) } }{ \sqrt{ { \sin}^(2) x} } }} \\ \\ = { \rm{ (1 + \cos(x) )/( \sin(x) ) }} \\ \\ = { \rm{ (1)/( \sin(x) ) + ( \cos(x) )/( \sin(x) ) }} \\ \\ = { \boxed{ \rm{ \: \csc(x) + \cot(x) \: }}}`