Question

How do you prove `cot^2 x/ (1+csc x) = (1-sin x)/ sin x`?

Answer 1

As proved below.

Explanation:

`cot^2 x = csc^2 x - 1, csc x = 1/ sin x`

`cot^2 x / (1 + csc x) = (csc^2 x - 1) / (csc x + 1)`

`=> (cancel(csc x + 1) (csc x - 1)) / cancel(csc x + 1)`

`=> csc x - 1 = (1/sin x - 1) color(violet)(= (1-sin x) / sin x`