Question

докажите торжество
arctg(1/2)+arcctg(1/3)=pi/4

Answer 1

task/26323516
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Докажите  тождество :
arctg(1/2) + arcctg(1/3) = π /4 
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 Пусть   arctg(1/2) + arcctg(1/3)= φ  
* * *  tg(α + β) = ( tgα + tgβ) / (1 -tgα *tgβ )  * * *
tgφ = tg( arctg1/2 + arcctg1/3 ) =tg( arctg1/2 + arctg3 )
( tg( arctg1/2)  +tg( arctg3) ) / (1 - tg( arctg1/2) *tg( arcctg3 ) ) =
= (1/2 +3) /(1 -(1/2)*3) =3,5 /(-0,5) = - 7 .   φ ≠ π/4 ⇒ пример неверно .
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Верное  тождество :   arctg(1/2) + arctg(1/3) = π /4 .
 φ = arctg1/2 + arctg1/3  
tgφ = tg( arctg1/2 + arctg1/3 ) =
( tg( arctg1/2  +tg( arctg1/3 ) / (1 -tg( arctg1/2) *tg( arctg1/3 ) ) =
= (1/2 +1/3) / ( 1 -(1/2)*(1/3) ) = (5 /6 )  / (5/6) = 1 .   ⇒   φ = π/4 .