Equivalence of two Ito formulae Let $X$ and $Y$ be two $1$-dimensional Ito processes. There are two Ito formulae for the product $X_tY_t$ given by 


*

*$d\left(X_tY_t\right)=X_tdY_t+Y_tdX_t+d\left[X_t,Y_t\right]$

*$d\left(X_tY_t\right)=X_tdY_t+Y_tdX_t+dX_tdY_t$.


$dX_tdY_t$ is to be calculated using the rules $dt\cdot dt=0$, $dt\cdot dW_t=0$, $dW_t\cdot dW_t=dt$. Clearly for the above to be consistent we need $dX_tdY_t=d\left[X_t,Y_t\right]$ but I am unable to see why this holds. Would anyone be able to shed some light on this?
Edit: Fixed formulae.
 A: First of all the product formulae which you wrote are incorrect, it should have been 
$$ d(X_tY_t) = X_t \mathrm{d}Y_t + Y_t\mathrm{d}X_t + \mathrm{d}X_t\mathrm{d}Y_t.$$
However, there is a lot of notation involved in the above. Let me start without the abuse of notation, for two Ito processes $X$ and $Y$ we have that
$$( \star) \ \qquad  X_tY_t = X_0Y_0 + \int_0^t X_s \mathrm{d}Y_s + \int_0^t Y_s \mathrm{d}X_s +[X,Y]_t,$$
where $[X, Y]$ stands for a covariation of two processes (link).
We can re-writte the equation $( \star) $ as follows
$$\int_0^t \mathrm{d}(X_sY_s) = \int_0^t X_s \mathrm{d}Y_s + \int_0^t Y_s \mathrm{d}X_s +\int_0^t \mathrm{d}[X,Y]_s.$$
Now we make use of the following abuse of notation
$$ \mathrm{d}(XY) = X\mathrm{d}Y + Y\mathrm{d}X + \mathrm{d}X\mathrm{d}Y,$$
where $\mathrm{d}X\mathrm{d}Y$ denotes $\mathrm{d}[X,Y]$. This notation is a brilliant one because it make our calculations quicker and it still easy to come back to the formal ones. In particular, it is widely used in quantitative finance. Here are some notes where you can find properties of the quadratic variation for Ito processes (without proofs) link.
