Precalculus, Chapter 5, 5.2, Section 5.2, Problem 18

You need to remember that 1/(csc t) = sin t , hence, replacing sin t for 1/(csc t) to the left side, yields:
sin t*cot^3 t = (cos t)*(csc^2 t - 1)
You need to replace (cos^3 t)/(sin^3 t) for cot^3 t , to the left side:
sin t*(cos^3 t)/(sin^3 t) = (cos t)*(csc^2 t - 1)
Reducing by sin t to the left, yields:
(cos^3 t)/(sin^2 t) = (cos t)*(csc^2 t - 1)
Reducing by cos t both sides, yields:
(cos^2 t)/(sin^2 t) = (csc^2 t - 1)
cot^2 t = csc^2 t - 1
Replacing 1/(sin^2 t) for csc^2 t yields:
cot^2 t = 1/(sin^2 t) - 1 => cot^2 t + 1 = 1/(sin^2 t)
Notice that the identity cot^2 t + 1 = 1/(sin^2 t) is valid, since it is derived from sin^2 t + cos^2 t = 1 .
Hence, verifying if the given identity (cot^3 t)/(csc t) = (cos t)*(csc^2 t - 1) is valid, yields that it is.