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Консультация № 198281
17.04.2020, 12:01
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе: Доказать: arctgx+arcctgx=p/2 ,если x принадлежит R
Обсуждение
22.07.2022, 06:42
общий
22.07.2022, 06:45
это ответ
Здравствуйте, pilot!
Пусть a=arctg(x), b=arcctg(x). Тогда, с одной стороны, x=tg(a), где a[$8712$](-[$960$]/2; [$960$]/2); с другой стороны, x=ctg(b), где b[$8712$](0; [$960$]). Отсюда следует, что tg(a)=ctg(b), или tg(a)=tg([$960$]/2-b), причём ([$960$]/2-b)[$8712$](-[$960$]/2; [$960$]/2). Тогда a=[$960$]/2-b, или a+b=[$960$]/2, или arctg(x)+arcctg(x)=[$960$]/2, что и требовалось доказать.
Пусть a=arctg(x), b=arcctg(x). Тогда, с одной стороны, x=tg(a), где a[$8712$](-[$960$]/2; [$960$]/2); с другой стороны, x=ctg(b), где b[$8712$](0; [$960$]). Отсюда следует, что tg(a)=ctg(b), или tg(a)=tg([$960$]/2-b), причём ([$960$]/2-b)[$8712$](-[$960$]/2; [$960$]/2). Тогда a=[$960$]/2-b, или a+b=[$960$]/2, или arctg(x)+arcctg(x)=[$960$]/2, что и требовалось доказать.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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