Verify csc^4 - cot^4 = 2cot^2 + 1

you can factor the left side of the equation to

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

then, since cot^2 + 1 = csc^2, you can replace the csc^2 in the left parentheses with cot^2 + 1, and the cot^2 in the right parentheses with csc^2 - 1 (which equals cot^2)

then, youll end up with

(cot^2 + 1 - cot^2) (csc^2 + csc^2 - 1)

combine like terms, and the left parentheses becomes 1 (cot^2 - cot^2 = 0), and the right parentheses becomes 2csc^2 - 1.

1 (2csc^2 - 1) = 2csc^2 - 1