Help with pre-calculus?!?

Question

How would I prove this is 4 or more steps.

csc^2 Θ – cot^2 Θ = cot^2 Θ (1-cos^2 Θ)
sec^2 Θ


(csc -cot is all being divided by sec) (and you cant mess with the right side)

Answer 1

(cosec^2 Θ - cot^2 Θ) / (sec^2 Θ)
= (1/sin^2 Θ - cos^2 Θ / sin^2 Θ) / (1/cos^2 Θ)
= (cos^2 Θ / sin^2 Θ) (1 - cos^2 Θ)
= cot^2 Θ (1 - cos^2 Θ)

Of course, since 1 - cos^2 Θ = sin^2 Θ we can simplify this to cot^2 Θ sin^2 Θ = cos^2 Θ.

Answer 2

Split the left into 2 fractions and make replacements:

(1/sin^2 Θ) / (1/cos^2 Θ) - (cos^2 Θ/sin^2 Θ)/(1/cos^2 Θ)

flip and multiply to get

cos^2 Θ / sin^2 Θ - cos^4 Θ/ sin^2 Θ
= [cos^2 Θ - cos^4 Θ]/sin^2 Θ

Factor out a cos^2 Θ

cos^2 Θ (1 - cos^2 Θ) over sin^2 Θ
Replace cos^2 Θ / sin^2 Θ with cot^2 Θ and you have it