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已知a²+b²=1 证明:b/a+1 -a/b+1=2(b-a)/a+b+1
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已知a²+b²=1 证明:b/a+1 -a/b+1=2(b-a)/a+b+1
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目测是要证明b/(a+1)-a/(b+1) = 2(b-a)/(a+b+1).
b/(a+1)-a/(b+1)
= (b(b+1)-a(a+1))/((a+1)(b+1))
= (b-a)(a+b+1)/(1+a+b+ab)
= (b-a)(a+b+1)²/((1+a+b+ab)(a+b+1))
= (b-a)(a²+b²+1+2a+2b+2ab)/((1+a+b+ab)(a+b+1))
= (b-a)(2+2a+2b+2ab)/((1+a+b+ab)(a+b+1))
= 2(b-a)/(a+b+1).
b/(a+1)-a/(b+1)
= (b(b+1)-a(a+1))/((a+1)(b+1))
= (b-a)(a+b+1)/(1+a+b+ab)
= (b-a)(a+b+1)²/((1+a+b+ab)(a+b+1))
= (b-a)(a²+b²+1+2a+2b+2ab)/((1+a+b+ab)(a+b+1))
= (b-a)(2+2a+2b+2ab)/((1+a+b+ab)(a+b+1))
= 2(b-a)/(a+b+1).
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