Question

Решите уравнение

arctg(x+1) + arctg(x-1) = arctg2​

Answer 1

Пусть

arctg(x+1)=α;  ⇒  tgα=x+1;   α∈(-π/2;π/2)

arctg(x-1)=β  ⇒   tgβ=x-1;  β∈(-π/2;π/2)

Уравнение:

α+β=arctg2

tg(α+β)=tg(arctg2)

tg(α+β)=2

$\frac{tg\alpha +tg\beta }{1-tg\alpha\cdot tg\beta } =2$

$\frac{(x+1) +(x-1)}{1-(x+1)(x-1)} =2\\\\\frac{2x}{2-x^2}=2\\\\2x=4-2x^2\\\\x^2+x-2=0; \\\\x=1;x=-2$

при х=1

arctg2+arctg0=arctg2 - верно

при х=-2

arctg(-1)+arctg(-3)=arctg2 - неверно

О т в е т. 1