Задача 24687 1/tg(x/2) - 1/ctg(x/2) - 1 - 2ctgx =...
Условие
математика 10-11 класс
5084
Решение
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sin(x/2) ≠ 0⇒ (x/2)≠ Pin, x≠ 2Pin, n ∈ Z
cos(x/2) ≠ 0⇒ (x/2)≠(Pi/2)+ Pim, x≠ Pi+2Pim, m ∈ Z
1/tg(x/2)=ctg(x/2)=cos(x/2)/sin(x/2);
1/ctg(x/2)=tg(x/2)=sin(x/2)/cos(x/2);
1/tg(x/2) - 1/ctg(x/2)=
=cos(x/2)/sin(x/2)- sin(x/2)/cos(x/2)=
=(cos^2(x/2)-sin^2(x/2))/sin(x/2)*cos(x/2)=
=2cosx/sinx=2ctgx
2ctgx-1-2ctgx=sin2x
sin2x=-1
2x=(-Pi/2)+2Pik, k ∈ Z
x=(-Pi/4)+Pik, k ∈ Z
О т в е т. (-Pi/4)+Pik, k ∈ Z