Reciprocal trick

 It’s just an algebraic identity (valid for 

x≠0,−1x\neq 0,-1) that comes from the symmetry x↔1xx \leftrightarrow \tfrac1x:

11+x+11+1x=11+x+xx+1=1+xx+1=1.\frac{1}{1+x}+\frac{1}{1+\frac{1}{x}} =\frac{1}{1+x}+\frac{x}{x+1} =\frac{1+x}{x+1}=1.

Many contest solutions use it as the “reciprocal trick” or note it as the identity f(x)+f(1/x)=1f(x)+f(1/x)=1 for f(t)=11+tf(t)=\frac{1}{1+t}.

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