Question

Y=(tgx+ctgx) (cos2x+ctg2x)

Answer 1

производная произведения, далее на фото

Answer 2

Y=(tgx+ctgx) (cos2x+ctg2x)
tg(x)=t
Y=(tgx+ctgx) (cos2x+ctg2x)=(t^2+1)/t * ((1-t^2)/(1+t^2)+(1-t^2)/(2t)) =
=(t^2+1)(1-t^2)/t * (1/(1+t^2)+1/(2t)) =(t^2+1)(1-t^2)/t *(1+t)^2 /((1+t^2)(2t)) =
=(1-t^2) *(1+t)^2/(2t^2)
y`=dy/dt *t`={ (-2t *(1+t)^2+2(1-t^2) *(1+t)) *(2t^2) - (1-t^2) *(1+t)^2*4t }*(1+t^2) / 4t^4 =
={ (- t *(1+t)^2+(1-t^2) *(1+t)) * t - (1-t^2) *(1+t)^2 }*(1+t^2) / t^3 =
={ (- t *(1+t)+(1-t^2)) * t - (1-t^2) *(1+t) }*(1+t^2)(1+t) / t^3 =
={ (1-2t) * t - (1-t^2) }*(1+t^2)(1+t)^2 / t^3 =
={ t-2t^2 -1+t^2) }*(1+t^2)(1+t)^2 / t^3 =
=( -t^2+t-1)*(1+t^2)(1+t)^2 / t^3 = ( -t^2+t-1)*(1+t)^2 cos(x) / sin^3(x) =
( tg(x)-1/cos^2(x))*(1+tg(x))^2 cos(x) / sin^3(x) = (sin(x)-1/cos(x))*(1+tg(x))^2 / sin^3(x)