cot²x /(cscx + 1) = (1 - sinx) /sinx
let's apply the identity cot²x = csc²x - 1:
(csc²x - 1) /(cscx + 1) = (1 - sinx) /sinx
(factoring csc²x - 1 as a difference of squares and then simplifying)
[(cscx + 1)(cscx - 1)] /(cscx + 1) = (1 - sinx) /sinx
cscx - 1 = (1 - sinx) /sinx
let's recall that cscx = 1 /sinx:
(1 /sinx) - 1 = (1 - sinx) /sinx
ending with:
(1 - sinx) /sinx = (1 - sinx) /sinx (verified)
I hope it helps
