<s>[INST] How to simplify $\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$? $$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$
I have been staring at it for ages and know that it simplifies to $x$, but have been unable to make any significant progress.
I have tried doing $(\frac{1-x}{1-2x})(\frac{1+2x}{1+2x})$ but that doesn't help as it leaves $\frac{1+x-2x^2}{1-4x^2}$ any ideas? 
 [/INST] $$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$
(take L.C.M. in numerator and denominator)
$$\dfrac{\dfrac{{(1-2x)}-{(1-x)}}{1-2x}}{\dfrac{(1-2x)-2{(1-x)}}{1-2x}}$$
$$\dfrac{\dfrac{1-2x-1+x}{1-2x}}{\dfrac{1-2x-2+2x}{1-2x}}$$
$$\dfrac{\dfrac{-x}{1-2x}}{\dfrac{-1}{1-2x}}$$
since $$\dfrac{\dfrac ab}{\dfrac cd}=\dfrac ab\times\dfrac {d}{c}$$
so
$${\dfrac{-x}{1-2x}}\times{\dfrac{1-2x}{-1}}$$
$$x$$
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