Доказать тождество:sin^3t(1+ctgt)+cos^3t(1+tgt)=sint+cost

Решите задачу:

$sin^3t(1+ctgt)+cos^3t(1+tgt)=sint+cost \\ \\ sin^3t(1+ctgt)+cos^3t(1+tgt)=sin^3t(1+\frac{cost}{sint})+cos^3t(1+\frac{sint}{cost})= \\ \\ =sin^3t+sin^3t*\frac{cost}{sint}+cos^3t+cos^3t*\frac{sint}{cost}=sin^3t+sin^2tcost+cos^3t+cos^2tsint= \\ \\ =(sin^3t+cos^3t)+sintcost(sint+cost)=(sint+cost)(sin^2t-sintcost+ \\ \\ +cos^2t)+sintcost(sint+cost)=(sint+cost)(sin^2t-sintcost+cos^2t+ \\ \\ +sintcost)=(sint+cost)(sin^2t+cos^2t)=(sint+cost)*1=sint+cost$