Question

Ctg(13π/12)-ctg(15π/12)

Answer 1

ctg(13π/12)-ctg(15π/12)= ctg π/12 - ctg 3π/12= ctg15 -ctg45= tg 15- tg 45= tg 15- √$\frac22$

Answer 2

Для начала, давайте выразим ctg(13π/12) и ctg(15π/12) через синусы и косинусы, используя определения:

ctg(13π/12) = 1/tan(13π/12) = 1/(sin(13π/12)/cos(13π/12)) = cos(13π/12)/sin(13π/12)

ctg(15π/12) = 1/tan(15π/12) = 1/(sin(15π/12)/cos(15π/12)) = cos(15π/12)/sin(15π/12)

Теперь посмотрим на значения cos(13π/12) и cos(15π/12), а также sin(13π/12) и sin(15π/12).

cos(13π/12) = cos(π/6 + π) = cos(π/6) = √3/2

cos(15π/12) = cos(π/4 + π) = cos(π/4) = √2/2

sin(13π/12) = sin(π/6 + π) = sin(π/6) = 1/2

sin(15π/12) = sin(π/4 + π) = sin(π/4) = √2/2

Теперь подставим значения в выражение ctg(13π/12) - ctg(15π/12):

ctg(13π/12) - ctg(15π/12) = (cos(13π/12)/sin(13π/12)) - (cos(15π/12)/sin(15π/12))

= ((√3/2)/(1/2)) - ((√2/2)/(√2/2))

= (√3/2) * 2 - 1

= √3 - 1

Таким образом, ctg(13π/12) - ctg(15π/12) = √3 - 1.