On the definition of the Zariski Tangent space I have the following (relating to the definition of the Zariski tangent space) in my notes. 
Let $m_P$ be the ideal of $P$ in $k[V]$, and $M_P$ as the ideal $\langle X_1,...,X_n\rangle\subset k[X_1,...,X_n]$. Then $m_P=M_P/I(V)$. Then 
$$m_P/m_P^2=M_P/(M_P^2+I(V))$$
I'm having a lot of trouble seeing the above equality. I haven't really got anywhere past the obvious
$$m_P/m_P^2=(M_P/I(V))/(M_P/I(V))^2$$
Presumably $m_P^2=(M_P/I(V))^2=(M_P^2+I(V))/I(V)$ which would yield the desired equality but I just can't see it. We know that $m_P^2$ is the product of the ideal $m_P$ with itself. That is the ideal generated by polynomials $f\cdot g $ where $f$ and $g$ are in $M_P/I(V)$. I'm getting very confused with products of quotients etc... 
 A: $m_P^2$ is $(M_P^2 + I(V)) / I(V)$, so your explanation is right.
More generally, if $R$ is a ring and $I, J$ are ideals of $R$ with $I \subset J \subset R$, then the ideal $(J/I)^2$ of the quotient ring $R / I$ is equal to the ideal $(J^2 + I)/I$.
To see this, observe first of all that any ideal $\bar K$ of $R / I$ is of the form $K / I$ for some ideal $K$ of $R$ such that $I \subset K \subset R$; this $K$ contains precisely those elements in $R$ whose residue classes modulo $I$ are contained in $\bar K$. Applying this to our scenario, if $\bar K$ is $(J / I)^2$, then $\bar K = K / I$, where $K$ contains precisely those elements in $R$ that are congruent modulo $I$ to some element of $J^2$. In other words, $K = J^2 + I$, so $(J/I)^2 = (J^2 + I)/I$.
A: Let $R$ be a ring, $J$ an ideal of $R$, and $\pi$ be the canonical projection $R \to R/J$.
Consider an ideal $I$ of $R$. I find $\pi(I)$ — the ideal of $R/J$ generated by the image of $I$ — to be a much more convenient notion than its explicit construction va cosets $(I+J)/J$.
(this is related to the fact that, for $a \in R$, I find $\pi(a)$ a much more convenient notion to work with than $a + J$ is)
For example, in my opinion the path to seeing that $\pi(I)^2 = \pi(I^2)$ is rather direct, if not outright obvious. Trying to do that with the formula via cosets, however, introduces extra technical details that get in the way.
