Question

ВЫЧИСЛИТЕ : ctg п/8-tg п/8

Answer 1

Распишем это как (cos П/8)/(sin П/8)-(sinП/8)/(cosП/8)= приведем к общему знаменателю получим (cos^2П/8-sin^2П/8)/sinП/8*cosП/8= cosП/4/sinП/4=ctg П/4 = 1

Answer 2

Используем формулы тангенса и котангенса в смежных углах:

tg(π/2 - α) = 1/ tg α
ctg(π/2 - α) = 1/ ctg α

Значит, ctg(π/8) = 1/ tg(π/8)

tg(π/4 - π/8) = tg(π/8)
tg(π/4 - π/8) = (tg π/4 - tg π/8)/(1 + tg π/4 * tg π/8)
(1 - tg π/8)/(1 + tg π/8) = tg(π/8)

Таким образом, ctg(π/8) - tg(π/8) = 1/(tgπ/8) - tg(π/8) =
= (1 - tg²(π/8))/(tg(π/8)) = (1 - sin²(π/8))/(sin(π/8)/cos(π/8)) =
= (cos²(π/8))/(sin(π/8)) = cos(π/4)/sin(π/8) = √2/(2sin(π/8))

Ответ: √2/(2sin(π/8))