Please make me understand this

to prove sqrt( ( 1+cos(x)) / ( 1 - cos(x) ) )  = cosec(x) + cot(x)

LHS
( 1+cos(x) )  / ( 1 - cos(x) )
= [ ( 1+cos(x) )( 1+cos(x) )]  /  [( 1 - cos(x) )  ( 1+cos(x) ) ]  # multiply numerator and denominator by ( 1+cos(x) )
= ( 1+cos(x) )^2/(1 -cos^2(x)
=  ( 1+cos(x) )^2/sin^2(x)
thus
( 1+cos(x) )  / ( 1 - cos(x) )  =   ( 1+cos(x) )^2/sin^2(x)
sqrt(  ( 1+cos(x) )  / ( 1 - cos(x)  ) = sqrt ( ( 1+cos(x) )^2/sin^2(x)  ) = (1 + cos(x) )/sin(x) = cosec(x) + cot(x)