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

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

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