cot^3(T)/ csc(T)= cos(T)(csc^2(T)-1)?

Question

proved the identity is true

Answer 1

I'm just going to manipulate the left until it looks like the right:

cot^3 (T) / csc (T)
cot^3 (T) * 1/csc (T)
cot^3 (T) * sin T
[cos^3 T / sin^3 T] * sin T
cos^3 T / sin^2 T
cos T * cos^2 T / sin^2 T
cos T * cot^2 T
cos T * (csc^2 T - 1)
---------
Note: cot^2 (T) = csc^2 (T) - 1
because:
sin^2 T + cos^2 T = 1
(divide both sides by sin^2 T)
1 + (cos^2 T / sin^2 T) = 1/sin^2 T
1 + cot^2 T = csc^2 T

Answer 2

cot³ T/cosecT = (cos³ T/sin³ T) / cosecT

= (cos³ T / sin³ T) sinT

= cosT(cos² T)(sinT) / (sin² T)(sinT)

= cosT cos² T / sin² T = cosTcot² T

= cosT(cos² T / sin² T) = cosT(1 - sin² T) / sin² T

= cosT(1 / sin² T - 1) = cosT(cosec² T - 1) as required.