Задание: докажите,что..
[tex]sin ^{3} \alpha (1+ctg \alpha )+cos ^{3} \alpha (1+ tg \alpha )=sin \alpha +cos \alpha [/tex]
[tex]sin ^{3} \alpha (1+ctg \alpha )+cos ^{3} \alpha (1+ tg \alpha )=sin \alpha +cos \alpha [/tex]
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Verified answer
Sin³α(1+ctgα)+cos³α(1+tgα) ==sin³α(1 + cosα) +cos³α(1 + sinα) =
sinα cosα
=sin³α(sinα + cosα) + cos³α (cosα + sinα) =
sinα cosα
= sin²α (sinα + cosα) + cos²α (cosα + sinα) =
=(sinα + cosα) (sin²α + cos²α) =
= (sinα + cosα) * 1 =
= sinα + cosα
sinα + cosα = sinα + cosα, что и требовалось доказать.
Verified answer
Sin³α(1+ctqα) +cos³α(1+tqα) = sin³α+cos³α +sin²αcosα +cos²αsinα =sin³α+cos³α +3sin²αcosα +3cos²αsinα -2(sin²αcosα +cos²αsinα) =
(sincα+cosα)³ -2sinαcosα(sinα+cosα) =(sinα+cosα) ((sinα+cosα)² -2sinαcosα) =
(sinα+cosα) (sin²α+2sinαcosα+cos²α -2sinαcosα) =(sinα+cosα) (sin²α+cos²α)=
sinα+cosα .
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sin³α+cos³α +sin²αcosα +cos²αsinα =(sinα+cosα)³ -3sinαcosα(sinα+cosα) +sinαcosα(sinα+cosα) = (sinα+cosα)³ -2sinαcosα(sinα +cosα) =(sinα +cosα)((sinα +cosα)² -2sinαcosα) = ...