Question

Докажите тождество tga+ctgb/ ctga+tgb = tga*ctgb

Answer 1

$\frac{tg\alpha+ctg\beta}{ctg\alpha+tg\beta}=\frac{\frac{sin\alpha}{cos\alpha}+\frac{cos\beta}{sin\beta}}{\frac{cos\alpha}{sin\alpha}+\frac{sin\beta}{cos\beta}}=\frac{\frac{sin\alpha sin\beta+cos\beta cos\alpha}{cos\alpha sin\beta}}{\frac{cos\alpha cos\beta+sin\beta sin\alpha}{sin\alpha cos\beta}}=\frac{\frac{1}{cos\alpha sin\beta}}{\frac{1}{sin\alpha cos\beta}}=\frac{sin\alpha cos\beta}{cos\alpha sin\beta}=tg\alpha ctg\beta$