Question

Даю 30 баллов
Найдите cos два альфа, если ctg альфа = корень из 2 - 1

Answer 1

$Ctgalpha=sqrt{2}-1\\ Ctg^{2}alpha=(sqrt{2}-1)^{2}=2-2sqrt{2}+1=3-2sqrt{2}\\tg^{2}alpha=frac{1}{3-2sqrt{2} }\\tg^{2} alpha+1=frac{1}{Cos^{2}alpha}\\Cos^{2} alpha =frac{1}{1+tg^{2}alpha}=frac{1}{1+3-2sqrt{2} }=frac{1}{4-2sqrt{2} }\\Cos2alpha =2Cos^{2}alpha-1=2*frac{1}{4-2sqrt{2} }-1=frac{1}{2-sqrt{2} } -1=frac{1-2+sqrt{2} }{2-sqrt{2} }=-frac{sqrt{2}-1 }{sqrt{2}(sqrt{2}-1)}=-frac{sqrt{2} }{2}$

Второй способ :

$frac{Sin^{2}alpha}{Sin^{2}alpha}}=frac{Ctg^{2}alpha-1}{Ctg^{2}alpha+1}=frac{(sqrt{2}-1)^{2}-1}{(sqrt{2}-1)^{2}+1}=frac{2-2sqrt{2}+1-1 }{2-2sqrt{2}+1+1 } =frac{2-2sqrt{2} }{4-2sqrt{2} }=frac{2(1-sqrt{2}) }{2(2-sqrt{2}) }=$$frac{1-sqrt{2} }{sqrt{2}(sqrt{2}-1)}=-frac{sqrt{2} }{2}$