<s>[INST] Prove that$\frac{x^{2^{k-1}}}{\left(1-x^{2^{k}}\right)}$= $\frac{1}{1-x^{2^{k-1}}}-\frac{1}{1-x^{2^{k}}}$ QuestionProve that$\frac{x^{2^{k-1}}}{\left(1-x^{2^{k}}\right)}$=
$\frac{1}{1-x^{2^{k-1}}}-\frac{1}{1-x^{2^{k}}}$
My Approach R.H.S 
$\frac{1}{1-x^{2^{k-1}}}-\frac{1}{1-x^{2^{k}}}$=$\frac{x^{2^{k-1}\left(1-x^{2}\right)}}{\left(1-x^{2^{k}}\right)\left(1-x^{2^{k-1}}\right)}$ 
i just don't know what else i can do?
 [/INST] Consider $x^{2^k}=m$
The equation reduces to 
$$\frac {\sqrt m}{1-m}=\frac {1+\sqrt m -1}{1-m}$$
$$=\frac {1+\sqrt m}{1-m}-\frac {1}{1-m}$$
$$=\frac {1}{1-\sqrt m} - \frac {1}{1-m}$$
Resubstitute value of $m$ to obtain desired proof
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