Immediate help with proving identities in trig

Need help verfying... csc(-x)-sin(-x)=-cosxcotx
sin^2x(1+cot^2x)=1

For the first identity:
From a knowledge of the Law of Quadrants or by testing on your calculator
sin (-x) = - sin (x)
cosec (-x) = 1 / [sin (-x)]
cosec (-x) = -1 / sin x
Thereofre starting with the left-hand side
cosec (-x) - sin(-x) = (-1 / sin x) + sin x
Put these over a common denominator of sin x
cosec (-x) - sin (-x) = [(-1 +sin² x) / sin x]
cosec (-x) - sin (-x) = - cos² x / sin x
cosec (-x) - sin (-x) = - cos x . cos x / sin x
cosec (-x) - sin (-x) = - cos x cot x
= Right-hand side

For the second identity
sin² x (1 + cot² x) = sin² x (cosec² x)
sin² x (1 + cot² x) = sin² x (1 / sin² x)
= 1 = Right-hand side