<s>[INST] Simplyfying the equation 
It is given that, $a+b+c=0$ we will have to prove that, 
  $$\frac{1}{x^b + x^{-c}+1} + \frac{1}{x^a + x^{-b}+1} + \frac{1}{x^c +
 x^{-a}+1} =1$$

What I have done is, $a=-(b+c)$ into the equation and also I have tried with multiplying $x^a, x^b x^c$ but it didn't look simple. What should be done here?
 [/INST] Keeping the first term as is, I am trying to set the numerator of the rest two terms as parts of the denominator of the first term.
$$\frac1{x^a+x^{-c}+1}=\frac{x^b}{x^{b+a}+1+x^b}=\frac{x^b}{x^{-c}+1+x^b}\text{ as }b+a=-c$$
$$\frac1{x^c + x^{-a}+1}=\frac{x^{-c}}{1+x^{-(c+a)}+x^{-c}}=\frac{x^{-c}}{1+x^b+x^{-c}}\text{ as }c+a=-b$$
</s>