How do you simplify (csc² t - cot² t) / sin² t?

1 Answer
Jake M.
Mar 4, 2018

sec^2 t

Explanation:

I like just sines and cosines, so let's work with those:

(csc^2 t - cot^2 t)/(sin^2(t))= (1/(sin^2t) - (cos^2t)/(sin^2t))/(sin^2t)

Let's get rid of the fractions up top by multiplying top and bottom by sin^2t

= (1 - cos^2t)/(sin^4 t)

Because sin^2t + cos^2 t = 1, 1- cos^2 t = sin^2 t.
= (sin^2t)/(sin^4t) = 1/(sin^2t) = sec^2 t

In hindsight, we could have also realized that there is a trig identity that
csc^2t-cot^2t = 1
but that involves memorization, whereas the first thing doesn't.

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