Упростите выражение ctgt+tg(-t)/cost+sin(-t)+tg(-t)/sint

Решите задачу:

$\dfrac{ctgt+tg(-t)}{cost+sin(-t)}+ \dfrac{tg(-t)}{sint} = \\ \\ = \dfrac{ \frac{cost}{sint}- \frac{sint}{cost} }{cost-sint}- \dfrac{ \frac{sint}{cost} }{sint} = \\ \\ = \dfrac{ \frac{cos^2t-sin^2t}{sintcost} }{cost-sint}- \dfrac{1}{cost} = \\ \\ = \dfrac{cos^2t-sin^2t}{sintcost(cost-sint)} - \dfrac{1}{cost}= \\ \\ = \dfrac{(cost-sint)(cost+sint)}{sintcost(cost-sint)}- \dfrac{1}{cost}= \\ \\ = \dfrac{cost+sint-sint}{sintcost}= \\ \\ = \dfrac{cost}{sintcost}= \\ \\ = \dfrac{1}{sint}$