<s>[INST] If $abc+1=0$, verify that' If $abc+1=0$, prove that:
$\frac {1}{1-a-b^{-1}}+\frac {1}{1-b-c^{-1}} +\frac {1}{1-c-a^{-1}}=1$.
My Attempt:
$abc+1=0$
$abc=-1$.
Now,
$$L.H.S=\frac {1}{1-a-b^{-1}}+\frac {1}{1-b-c^{-1}}+\frac {1}{1-c-a^{-1}}$$
$$=\frac{b}{b-ab-1} + \frac {1}{1-b-c^{-1}} + \frac {c^{-1}}{c^{-1}-1-(ca)^{-1}}$$
$$=\frac{b}{b+c^{-1} -1} +\frac{1}{1-b-c^{-1}} +\frac{c^{-1}}{c^{-1}-1+b}$$
What should I do next?
 [/INST] You are on the correct path. Your steps are all correct. Proceeding from there, just note that 
$$\frac{b}{b+c^{-1} -1} +\frac{1}{1-b-c^{-1}} +\frac{c^{-1}}{c^{-1}-1+b}$$
$$=\frac{b+c^{-1}}{b+c^{-1} -1} -\frac{1}{b+c^{-1}-1}$$
$$=\frac{b+c^{-1}-1}{b+c^{-1} -1}$$
$$=1$$
Hope this helps you.
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