Form an identity from the following expression: (csc^4x - 2csc^2x + 1)/(cot^2x) ?

Question

I have been at this forever, please help!!

Answer 1

Let's see.  You can factor the top and get

(csc^2(x) - 1)^2 / cot^2(x)

Since csc^2(x) - 1 = cot^2(x), this whole expression becomes

cot^2(x).

Answer 2

(csc^4x - 2csc^2x + 1)/(cot^2x)

=(csc^4(x) - 2csc^2(x) + 1)/(cot^2(x))

=(csc^2(x) - 1)^2  /  (cot^2(x)):trig identity 1+ cot^2(x) = csc^2(x)

=(cot^2(x))^2  /  (cot^2(x))

=cot^2(x) * cot^2(x) / cot^2(x)

=cot^2(x)

(csc^4x - 2csc^2x + 1)/(cot^2x) = cot^2(x)