Question

Verify the identity.
cot^3t/csct = cost(csc^2t-1)

Answer 1

First, turn everything into sin and cos!

Answer 2

OK
cot = cos/sin, so cot^3 = cos^3 / sin ^3
csc = 1/sin so 1/csc = sin

Left side:
cot^3t/csct
=(cos^3t / sin^3t) * sint
=cos^3t / sin^2t

Right side:
1+cot^2 = csc^2
csc^2 -1 = cot^2

so, right side:
cost(csc^2t-1)
=cost(cot^2t)
=cost( cos^2t/sin^2t)
=cos^3t/sin^2t

Both sides are identical!