solve the identity csc x - cot x = sin x / 1+cos x Step by step instructions please. 

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You need to remember that `csc x = 1/ sin x`  and `cot x = cos x/sin x` , hence, substituting `1/sin x`  for `cscx`  and  `cos x/sin x`  for `cot x,`  to the left side, yields:

`1/ sin x - cos x/sin x = (sin x)/(1 + cos x)`

`(1 - cos x)/sin x= (sin x)/(1 + cos x)`

Performing diagonal multiplication yields:

`sin^2 x = (1 - cos x)(1 + cos x)`

Converting the product into a difference of squares yields:

`sin^2 x = 1 - cos^2 x`

Moving the term `cos^2 x`  to the left side yields:

`sin^2 x + cos^2 x = 1`

Notice that solving the expression you have arrived to the fundamental formula of trigonometry, hence, the given expression represents the equivalent of fundamental formula `sin^2 x + cos^2 x = 1` , hence `csc x - cot x = (sin x)/(1 + cos x)`  holds.

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