tan x /(1-cot x) + cot x /(1-tanx) = 1+ sec x csc x求证!

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tan x /(1-cot x) + cot x /(1-tanx) = 1+ sec x csc x
求证!

tanx/(1 - cotx) + cotx/(1 - tanx)
= tanx/(1 - cosx/sinx) + cotx/(1 - sinx/cosx)
= sinxtanx/(sinx - cosx) + cosxcotx/(cosx - sinx)
= (sinxtanx - cosxcotx)/(sinx - cosx)
= (sin²x/cosx - cos²x/sinx)/(sinx - cosx)
= (sin³x - cos³x)/[sinxcosx(sinx - cosx)]
= [(sinx - cosx)(sin²x + sinxcosx + cos²x)]/[sinxcosx(sinx - cosx)]
= (1 + sinxcosx)/(sinxcosx)
= 1 + secxcscx

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