Question

докажите тождество$( frac{cos t}{tg t} + frac{sin t}{ctg t} ) : ( tg t + ctg t - 1) = sin t + cos t$

Answer 1

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