Question

Упростить выражения 1)1+tgB*tg2B 2) ctgL-ctg2L

Answer 1

$1) 1+tg \beta *tg2 \beta =1+ \frac{sin \beta }{cos \beta } * \frac{sin2 \beta }{cos2 \beta } =1+\frac{sin \beta }{cos \beta }* \frac{2sin \beta cos \beta }{cos ^{2} \beta -sin ^{2} \beta } =$

$=1+ \frac{2sin ^{2} \beta }{cos ^{2} \beta -sin ^{2} \beta } = \frac{cos ^{2} \beta -sin ^{2} \beta+2sin ^{2} \beta }{cos ^{2} \beta -sin ^{2} \beta} = \frac{cos ^{2} \beta +sin ^{2} \beta }{cos ^{2} \beta -sin ^{2} \beta} = \frac{1}{cos2 \beta }$

$2) ctg \alpha -ctg2 \alpha = \frac{sin(2 \alpha - \alpha )}{sin \alpha sin2 \alpha } = \frac{sin \alpha }{sn \alpha sin2 \alpha } = \frac{1}{sin2 \alpha }$