Question

I kind of know how to do these problems, but Im stuck on this one right now. Prove the formula below
$csc^{4}x - cot^{4}x = csc^{2}x + cot^{2}x$

Answer 1

Use difference of squares. (I'm leaving out the "x" part)

csc^4 - cot^4 = (csc^2 + cot^2)(csc^2 - cot^2)

From here, we should recognize one of the trig identities:

1 + cot^2 = csc^2
or rewritten as:
1 = csc^2-cot^2

So we can substitute the above into our previous equation:
(csc^2 + cot^2)(csc^2 - cot^2)
= (csc^2 + cot^2)(1)
= csc^2 + cot^2

And we have proven it. :)