1) (cos5α+cos6α+cos7α)/(sin5α+sin6α+sin7α); 2) tg(45˚+3α)-tg(45˚-3α).

Question

1) (cos5α+cos6α+cos7α)/(sin5α+sin6α+sin7α);
2) tg(45˚+3α)-tg(45˚-3α).

Answer 1

1
Воспользуемся формулами
cosa+cosb=2cos(a+b)/2*cos(a-b)/2
sina+sinb=2sin(a+b)/2*cos(a-b)/2
(2cos6acosa+cos6a)/(2sin6acosa+sin6a)=cos6a(2cosa+1)/[sin6a(2cosa+1)]=
=ctg6a
2
Воспользуемся формулами
tg(a+b)=(tga+tgb)/(1-tgatgb)
tg(a-b)=(tga-tgb)/(1+tgatgb)
(tg45+tg3a)/(1-tg45tg3a)-(tg45-tg3a)/(1+tg45tg3a)=
(1+tg3a)/(1-tg3a)-(1-tg3a)/(1+tg3a)=[(1+tg3a)²-(1-tg3a)²]/(1-tg²3a)=
=(1+tg3a-1+tg3a)(1+tg3a+1-tg3a)/(1-tg²3a)=4tg3a/(1-tg²3a)=2tg6a