Question

докажите тождество cos a/1+sin a - cos a/1-sin a = - 2tg a
даю 30 баллов!​

Answer 1

Решение:

$\frac{cos\alpha}{1+sin\alpha}-\frac{cos\alpha}{1-sin\alpha}=\frac{(1-sin\alpha)*cos\alpha-(1+sin\alpha)*cos\alpha}{(1+sin\alpha)*(1-sin\alpha)}=\frac{cos\alpha-sin\alpha*cos\alpha-(cos\alpha+sin\alpha*cos\alpha)}{1-sin^{2}\alpha}=\frac{cos\alpha-sin\alpha*cos\alpha-cos\alpha-sin\alpha*cos\alpha}{cos^{2}\alpha}=\frac{-(sin\alpha*cos\alpha+sin\alpha*cos\alpha)}{cos^{2}\alpha}=\frac{-sin(\alpha+\alpha)}{cos^{2}\alpha}=\frac{-sin2\alpha}{cos^{2}\alpha}=\frac{-2sin\alpha*cos\alpha}{cos^{2}\alpha}=$

$=\frac{-2sin\alpha}{cos\alpha}=-2*\frac{sin\alpha}{cos\alpha}=-2*tg\alpha=-2tg\alpha$

Ответ: Ч.Т.Д