Question

Упростите выражение. Срочно. Пожалуйста.
1) (1 + ctgL) / (1 + tgL)
2) tgL*ctgL - cos^2L
3) (1 - cosL)*(1 +cosL)
4) (cos*(2П+L)*tg(П+L) / (sin*(2П-L)*tg(П/2 +L)*ctg((3/2) П-L)
5) (cos(П+L)*tg((3/2)П+L) / (tg(П-L)*sin((3/2)П+L)
6) (1 - cos^2L + sin^2L) /(1 + cos^2L + sin^2L)

Answer 1

$frac{1+Ctg alpha }{1+tg alpha } = frac{1+ frac{1}{tg alpha } }{1+tg alpha } = frac{ frac{tg alpha +1}{tg alpha } }{1+tg alpha }= frac{tg alpha +1}{tg alpha (1+tg alpha )} = frac{1}{tg alpha }=Ctg alpha$

tgα * Ctgα - Cos²α = 1 - Cos²α = Sin²α

(1 - Cosα)(1 + Cosα) = 1 - Cos²α = Sin²α

$frac{Cos(2 pi + alpha )*tg( pi + alpha )}{Sin(2 pi - alpha )*tg ( frac{ pi }{2}+ alpha ) } *Ctg( frac{3 pi }{2}- alpha )= frac{Cos alpha *tg alpha }{-Sin alpha *(-Ctg alpha )} *tg alpha= frac{Cos alpha * frac{Sin alpha }{Cos alpha } }{Sin alpha * frac{Cos alpha }{Sin alpha } }*$$*tg alpha = frac{Sin alpha }{Cos alpha } *tg alpha =tg alpha *tg alpha =tg ^{2} alpha$

$frac{Cos( pi + alpha )*tg( frac{3 pi }{2}+ alpha ) }{tg( pi - alpha )*Sin( frac{3 pi }{2}+ alpha ) } = frac{-Cos alpha *(-Ctg alpha )}{-tg alpha *(-Cos alpha )}= frac{Ctg alpha }{tg alpha } = frac{ frac{1}{tg alpha } }{tg alpha }= frac{1}{tg ^{2} alpha } =$$=Ctg ^{2} alpha$

$frac{1-Cos ^{2} alpha +Sin ^{2} alpha }{1+Cos ^{2} alpha +Sin ^{2} alpha} = frac{Sin ^{2} alpha +Sin ^{2} alpha }{1+1}= frac{2Sin ^{2} alpha }{2}=Sin ^{2} alpha$