Задача 75635 ...

632f635587d19127bff140f0

1/31/2024 8:00:55 PM

Упростите выражение Ответы: 2tg α , 2/sin^(2) α , cos α /sin^(2) α , cos^(2) α

626fab9a354ab65fb2f43abb

1/31/2024 8:35:08 PM

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[m](tg \frac{9\pi}{4} + tg (\frac{5\pi}{2} - a))^2 + (ctg \frac{5\pi}{4} + ctg (\pi-a))^2 = A[/m]
Решим по действиям.
tg (9π/4) = tg (2π + π/4) = tg (π/4) = 1
ctg (5π/4) = ctg (π + π/4) = ctg (π/4) = 1
tg (5π/2 - a) = tg (2π + π/2 - a) = tg (π/2 - a) = ctg a
ctg (π - a) = ctg (-a) = -ctg a
Получаем:
A = (1 + ctg a)^2 + (1 - ctg a)^2 =
= 1 + 2ctg a + ctg^2 a + 1 - 2ctg a + ctg^2 a =
= 2 + 2ctg^2 a = 2*(1 + ctg^2 a) = 2/sin^2 a