Let $\mathfrak{a}$ be prime in $k[T_1,\dots,T_n]$ contained in $(T_1,\dots,T_n)$, with $k$ a field. Define the tangent cone $\text{TC}_x(X)$ of $X = V(\mathfrak{a})$ at $x=0 \in V(\mathfrak{a})$ to be the spectrum of \begin{align*} A = k[T_1,\dots,T_n] / \mathfrak{a}^* \end{align*} where $\mathfrak{a}^*$ is the homogeneous ideal generated by all the leading terms $f_r$ of all elements $f_r+f_{r+1} + \dots + f_n$ of $\mathfrak{a}$. Equivalently, it is the spectrum of \begin{align*} \bigoplus_{d \geq 0} \mathfrak{m}^d / \mathfrak{m}^{d+1} \end{align*} where $\mathfrak{m}$ is the maximal ideal of $x$. I'm trying to show that each irreducible component of $\text{TC}_x(X)$ has dimension $\dim X$, without using that $\dim \mathcal{O}_{X,x} = \dim \text{gr} \mathcal{O}_{X,x}$.
Let $W = \overline{ \{w\} }$ be an irreducible component of $\text{TC}_x(X)$. So I've tried to related chains of prime ideals \begin{align*} \mathfrak{p}_w = \mathfrak{p}_0 \subset \dots \subset \mathfrak{p}_k \end{align*} to chains of prime ideals that contain $\mathfrak{a}$, and conversely, both to no avail.
I know that if $\mathfrak{b}$ is any prime ideal, then the ideal generated by the homogeneous elements of $\mathfrak{b}$ is prime. This implies that $\mathfrak{p}_w$ is in fact homogeneous, and that we may assume that the above chain of prime ideals is homogeneous. I do not know how to use this.
Put $B = \mathcal{O}_X(X)$. I also know that we have a finite injective $k$-algebra homomorphism $k[T_1,\dots,T_d] \to B$, with $d = \dim X$. I think we can write $B$ then as $k[X_1,\dots,X_n]/\mathfrak{a}$, with $(X_{d+1} \dots, X_n) \subset \mathfrak{a}$. Then $A$ would be of the form $k[X_1,\dots,X_n]/\mathfrak{a}^*$ with $(X_{d+1} \dots, X_n) \subset \mathfrak{a}'$ as well. Is this a right direction?