Question

вычислить значение функции

Answer 1

$tgfrac{pi }{8}=sqrt{frac{sqrt2-1}{sqrt2+1}}\\frac{1+tg^2x+2tgx}{(1+sqrt2)tgx}=frac{(1+tgx)^2}{(1+sqrt2)tgx}\\(1+tgfrac{pi}{8})^2=(1+sqrt{frac{sqrt2-1}{sqrt2+1}})^2=(1+frac{sqrt{sqrt2-1}}{sqrt{sqrt2+1}})^2=\\=frac{(sqrt{sqrt2+1}+sqrt{sqrt2-1})^2}{sqrt2+1}=frac{sqrt2+1+2sqrt{(sqrt2-1)(sqrt2+1)}+sqrt2-1}{sqrt2+1}=\\=frac{2sqrt2+2sqrt{2-1}}{sqrt2+1}=frac{2sqrt2+2}{sqrt2+1}=frac{2(sqrt2+1)}{sqrt2+1}=2$

$(1+sqrt2)tgfrac{pi}{8}=(1+sqrt2)cdot frac{sqrt{sqrt2-1}}{sqrt{sqrt2+1}}=sqrt{sqrt2+1}cdot sqrt{sqrt2-1}=\\=sqrt{(sqrt2+1)(sqrt2-1)}=sqrt{(sqrt2)^2-1^2}=sqrt{2-1}=1\\\frac{1+tg^2frac{pi }{8}+2tgfrac{pi }{8}}{(1+sqrt2)tgfrac{pi }{8}}=frac{2}{1}=2$