Решите уравнениеarctg(x+1) + arctg(x-1) = arctg2. Created by Migma11. algebra-ru

Пустьarctg(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=arctg2tg(α+β)=tg(arctg2)tg(α+β)=2при х=1arctg2+arctg0=arctg2 - вернопри х=-2arctg(-1)+arctg(-3)=arctg2 - неверноО т в е т. 1

Решите уравнениеarctg(x+1) + arctg(x-1) = arctg2...

Migma11 @Migma11

August 2022 1 9 Report

Решите уравнение

arctg(x+1) + arctg(x-1) = arctg2

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nafanya2014

Verified answer

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

0 votes Thanks 2

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Answer

Answers & Comments

Verified answer

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