Question

ctg (a+pi) + 4 tg (3pi/2-a) + 2 tg (a+5pi/2) ctga = -0.25

Answer 1

$ctga + 4ctga+2tg(a+3 pi - frac{ pi }{2} )*ctga=-0.25$
$5ctga+2tg(a-frac{ pi }{2} )*ctga=-0.25$
$5ctga-2tg(frac{ pi }{2}-a)*ctga=-0.25$
$5ctga-2ctga*ctga=-0.25$
$-2ctg^{2}a+5ctga+0.25=0$
$2ctg^{2}a-5ctga-0.25=0$
$8ctg^{2}a-20ctga-1=0$

Замена: $ctga=t$, t∈R

$8t^{2}-20t-1=0, D=20^{2}+4*8=400+32=432$
$t_{1}= frac{20- sqrt{432}}{16} = frac{20-4 sqrt{27}}{16}= frac{5- sqrt{27}}{4}$
$t_{2}= frac{20+ sqrt{432}}{16} = frac{20+4 sqrt{27}}{16}= frac{5+ sqrt{27}}{4}$

Вернемся к замене:
1) $ctga=frac{5- sqrt{27}}{4}$
$a=arcctg(frac{5- sqrt{27}}{4})+ pi k$
2) $ctga=(frac{5+ sqrt{27}}{4}$
$a=arcctg(frac{5+ sqrt{27}}{4})+ pi k$
k∈Z

Answer 2

ctga=-0.25 отдельно