Question

Is (cotx + 1)^2 the same thing as cot^2 + 1? And from there, could I use the 1 + cot^2x = csc^2x identity to turn it into csc^2x ?

Answer 1

no they are not the same
--> (cotx +1)^2 = cot^2 +2cotx + 1

Answer 2

you could still use that identity to say that
(cotx +1)^2 = csc^2 + 2cotx