Докажите тождество[latex]( \frac{cos t}{tg t} + \frac{sin t}{ctg t} ) : ( tg t + ctg t - 1) = sin t + cos t[/latex]

[latex]( \frac{cost}{tgt}+ \frac{sint}{ctgt}):( \frac{sint}{cost} + \frac{cost}{sint}-1) = \\ \\ = ( \frac{cos^2t}{sint}+ \frac{sin^2t}{cost} ):( \frac{sin^2t+cos^2t-costsint}{cost*sint} )= \\ \\ = \frac{cos^3t+sin^3t}{sint*cost} * \frac{cost*sint}{sin^2t-costsint+cos^2t}= \\ \\ = \frac{(cost+sint)(sin^2t-costsint+cos^2t)}{sin^2t-costsint+cos^2t} =sint+cost[/latex]