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pavsasha
@pavsasha
July 2022
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Докажите тождество :
1/(1+ctga) - 1/(1-ctga) = tg 2a
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kirichekov
Verified answer
1/(1+ctgα)-1/(1-ctgα)=tg2α
1. 1+ctgα=1+cosα/sinα=(sinα+cosα)/sinα
1:((sinα+cosα)/sinα)=sinα/(sinα+cosα)
2. 1/(1-ctgα)=sinα/(sinα-cosα)
3. sinα/(sinα+cosα)-sinα/(sinα-cosα)=(sin²α-sinα*cosα-sinα²α-sinα*cosα)/((sinα+cosα)*(sinα-cosα))=(-2sinα*cosα)/(-(cos²α-sin²α))=sin2α/cos2α=tg2α
tg2α=tg2α
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Answers & Comments
Verified answer
1/(1+ctgα)-1/(1-ctgα)=tg2α1. 1+ctgα=1+cosα/sinα=(sinα+cosα)/sinα
1:((sinα+cosα)/sinα)=sinα/(sinα+cosα)
2. 1/(1-ctgα)=sinα/(sinα-cosα)
3. sinα/(sinα+cosα)-sinα/(sinα-cosα)=(sin²α-sinα*cosα-sinα²α-sinα*cosα)/((sinα+cosα)*(sinα-cosα))=(-2sinα*cosα)/(-(cos²α-sin²α))=sin2α/cos2α=tg2α
tg2α=tg2α