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  • Użytkownik Zadane
2009-10-10T22:00:48+02:00
A) tg²α-sin²α=tg²α*sin²α
L= tg²α-sin²α
P=tg²α*sin²α
L=tg²α-sin²α=(sinα/cosα)²-sin²α=(sin²α/cos²α)-sin²α=(sin²α/cos²α)-(sin²α*cos²α/cos²α)=(sin²α-sin²α*cos²α)/cos²α=[sin²α(1-cos²α)]/cos²α=(sin²α)*(1-cos²α/cos²α)=(sin²α)*(sin²α/cos²α)=(sin²α)*(sinα/cosα)²=sin²α*tg²α = P

b) cosα * ctgα + sinα = 1/sinα
L= cosα * ctgα + sinα
P=1/sinα
L=cosα*(cosα/sinα)+sinα=(cos²α/sinα)+sinα=(cos²α/sinα)+(sin²α/sinα)=(cos²α+sin²α)/sinα=1/sinα = P

c) cosα/(sinα-1) + tgα = -1/cosα
L=cosα/(sinα-1) + tgα
P=-1/cosα
L=cosα/(sinα-1) + tgα=
cosα/(sinα-1)+(sinα/cosα)=
(cosα*cosα)/[(sinα-1)*cosα] + (sinα*(sinα-1))/[(sinα-1)*cosα] = cos²α/[sinα*cosα-cosα] + (sin²α-sinα)/[sinα*cosα-cosα]=
[cos²α-sinα+sin²α]/[sinα*cosα-cosα]=
[1-sinα]/[cosα(sinα-1)]=
[(-1)(sinα-1)]/[cosα(sinα-1)]=
-1/cosα=P