Tg^2x+4tgx-5=0, -pi/2<x<0; ctg2x=?

Question

Tg^2x+4tgx-5=0, -pi/2<x<0; ctg2x=?

Answer 1

Tgx = a
a² + 4a - 5 = 0
D = 16 + 20 = 36
a = -5 или a = 1
tgx = -5                                          tgx = 1
x = -arctg 5 + πn                          x = π/4 + πk
x = - arctg 5  ∈ ( - π/2; 0)            x∉ ( - π/2 ; 0)

x = - arctg 5 , т.е. tg x = - 5  ⇒  ctg x  = - 1/5
ctg2x = (ctg²x - 1)/ (2 ctg x) = (1/25 - 1) / (-2/5) = 24/25 · 5/2 = 12/5