解下列問題 - Ans.fyi 參考答案

解下列問題

2007-01-27 8:48 pm

Prove the following identities

17) (tan^2 @ + tan @ + 1 )( cot^2 @ - cot@ +1)=tan^2@ + cot^2 @ + 1

18) tan@．1 - sin@/1 + cos@ = cot@．1 - cos@/ 1+ sin@

回答 (1)

myisland8132

2007-01-27 9:00 pm

✔ 最佳答案

17
LHS
=(tan^2 @ + tan @ + 1 )( cot^2 @ - cot@ +1)
=tan^2 @ ( cot^2 @ - cot@ +1)+tan @ ( cot^2 @ - cot@ +1)+( cot^2 @ - cot@ +1)
=1- tan@+tan^2@+cot@-1+tan@+cot^2 @ - cot@ +1
=tan^2@ + cot^2 @ + 1
=RHS
18
LHS
=tan@．(1 - sin@)/(1 + cos@)
=[sin@(1-sin@)]/[cos@(1+cos@)]
=[sin@(1-sin@)(1 - cos@)]/[cos@(1+cos@)(1 - cos@)]
=[sin@(1-sin@)(1 - cos@)]/[cos@sin^2@]
=[(1-sin@)(1 - cos@)]/[cos@sin@]
=[(1+ sin@ )(1-sin@)(1 - cos@)]/[(1+ sin@ )cos@sin@]
=[cos^2@(1 - cos@)]/[(1+ sin@ )cos@sin@]
=[cos@(1 - cos@)]/[(1+ sin@ )sin@]
=cot@．(1 - cos@)/( 1+ sin@)
=RHS

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