Question

How do you verify `cot^4 x+ cot^2x = csc^4 x - csc^2 x`?

Answer 1

It is

Write `csc^4 x` as `1/(sin^4 x)` and `cot^4 x` as `(cos^4 x)/(sin^4 x)`
`=> csc^4x - cot^4x = 1/(sin^4 x) - (cos^4 x)/(sin^4 x) = (1 - cos^4 x)/(sin^4 x)`

Now recall that `a^2 - b^2 = (a - b)*(a + b)` and use this fact with `a^2` being and
`b^2` being `cos^4 x` so that `a` is 1 and b is `cos^2 x`

So
`csc^4x - cot^4x = (1 - cos^4 x)/(sin^4 x) = (1 - cos^2 x)*(1 + cos^2 x) / sin^4 x`

But `cos^2 x + sin^2 x = 1` so that `1 - cos^2 x = sin^2 x`
so `csc^4x - cot^4x = sin^2 x * (1 + cos^2 x) / sin^4 x = (1 + cos^2 x) / sin^2 x = (1/sin^2 x) + (cos^2 x / sin^2 x) = csx^2 x + cos^2 x`