Question

доказать тождество по алгебре

Answer 1

$\displaystyle \tt sin^3\:\alpha(1+ctg\:\alpha)+cos^3\:\alpha(1+tg\:\alpha)=sin\:\alpha+cos\:\alpha\\\displaystyle \tt sin^3\:\alpha(1+\frac{cos\:\alpha}{sin\:\alpha})+cos^3\:\alpha(1+\frac{sin\:\alpha}{cos\:\alpha})=sin\:\alpha+cos\:\alpha\\\displaystyle \tt sin^3\:\alpha\cdot\frac{sin\:\alpha+cos\:\alpha}{sin\:\alpha}+cos^3\:\alpha\cdot\frac{cos\:\alpha+sin\:\alpha}{cos\:\alpha}=sin\:\alpha+cos\:\alpha$

$\displaystyle \tt sin^2\:\alpha\cdot(sin\:\alpha+cos\:\alpha)+cos^2\:\alpha\cdot(cos\:\alpha+sin\:\alpha)=sin\:\alpha+cos\:\alpha\\\displaystyle \tt (sin\:\alpha+cos\:\alpha)\cdot(sin^2\:\alpha+cos^2\:\alpha)=sin\:\alpha+cos\:\alpha\\\displaystyle \tt (sin\:\alpha+cos\:\alpha)\cdot1=sin\:\alpha+cos\:\alpha\\\displaystyle \tt sin\:\alpha+cos\:\alpha=sin\:\alpha+cos\:\alpha$