Question

помогите решить tg(10°)tg(80°)+ctg(10°)ctg(70°)​

Answer 1

$1); ; tg10cdot tg80+ctg10cdot ctg70=tg10cdot ctg10+frac{cos10cdot cos70}{sin10cdot sin70}=\\=1+frac{cos80+cos60}{cos60-cos80}=1+frac{frac{1}{2}+cos80}{frac{1}{2}-cos80}=1+frac{frac{1}{2}-1+(1+cos80)}{frac{1}{2}-1+(1-cos80)}=\\=1+frac{-frac{1}{2}+2cos^240}{-frac{1}{2}+2sin^240}=1+frac{4cos^240-1}{4sin^240-1}=frac{4sin^240-1+4cos^240-1}{4sin^240-1}=frac{2}{4sin^240-1}$

$2); ; tg10cdot tg80+ctg20cdot ctg70=tg10cdot ctg10+ctg20cdot tg20=1+1=2$

$P.S.; ; tga=ctg(90-a); ,; ; ctga=tg(90-a)\\tgacdot ctga=1$