解下列問題 - Ans.fyi 參考答案

解下列問題

2007-01-27 8:49 pm

Prove the following identities

19) ( 1- tan@)^2 + (1 - cot@)^2 - (sec@ - cosec@)^2 = 0

20) 1+ cos@ + sin@/1+ cos@ - sin@ = 1+ sin@/cos@

回答 (1)

myisland8132

2007-01-27 9:22 pm

✔ 最佳答案

19
L.H.S.
=(1-tanx)²+(1-cotx)²-(secx-cscx)²
=(1-2tanx+tan²x)+(1-2cotx+cot²x)-(sec²x-2secxcscx+csc²x)
=(-2tanx)+(-2cotx)+(2secxcscx)
=(-2sinx)/(cosx) + (-2cosx)/(sinx) + (2)/(sinxcosx)
=(2)/(sinxcosx) - (2)/(sinxcosx)
=0
=R.H.S.
20
LHS
=(1+cosx+sinx)/(1+cosx-sinx)
=(1+cosx+sinx)(1-(cosx-sinx))/(1+cosx-sinx)(1-(cosx-sinx))
=(1+cosx+sinx)(1-(cosx-sinx))/(1-(cosx-sinx)^2)
=(1+cosx+sinx)(1-cosx+sinx)/(2sinxcosx)
=(1-cosx+sinx+cosx-cos^2x+cosxsinx+sinx-sinxcosx+sin^2x)/(2sinxcosx)
=(1+2sinx-cos^2x+sin^2x)/(2sinxcosx)
=(2sinx+2sin^2x)/(2sinxcosx)
=(sinx+sin^2x)/(sinxcosx)
=(1+sinx)/cosx

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