<s>[INST] Prove using correlation to do t-test is equivalent to the standard t-test formula in linear regression? $\newcommand{\Cor}{\operatorname{Cor}} \newcommand{\Cov}{\operatorname{Cov}}$My question is why the following expression holds?
$$t = \frac{\hat{\beta_1}}{\operatorname{se}(\beta_1)} = \frac{\Cor(Y,X)\sqrt{n-2}}{\sqrt{1 - \Cor^2(Y,X)}}$$
Here is what I got so far:
\begin{align}
\frac{\hat{\beta_1}}{\operatorname{se}(\beta_1)} &= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\operatorname{se}(\beta_1)}\\
&= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\sqrt{\frac{\sum(\hat{Y_i} - Y_i)^2}{n-2}\cdot\frac{1}{\sum(X_i - \overline{X})^2}}}\\
&= \frac{\frac{\Cov(Y,X)}{S_x^2}\sqrt{n-2}}{\sqrt{\frac{\sum(\hat{Y_i} - Y_i)^2}{\sum(X_i - \overline{X})^2}}}\label{a}\tag{1}
\end{align} 
\begin{align}
\frac{\Cor(Y,X)\sqrt{n-2}}{\sqrt{1 - \Cor^2(Y,X)}}
&= \frac{\frac{\Cov(Y, X)}{S_xS_y}\sqrt{n-2}}{\sqrt{1 - \left(\frac{\Cov(Y, X)}{S_xS_y}\right)^2}} \label{b}\tag{2}
\end{align}
From $\ref{a}$ and $\ref{b}$, we then need to show:
\begin{align}
\frac{1}{S_x\sqrt{\frac{\sum(\hat{Y_i} - Y_i)^2}{\sum(X_i - \overline{X})^2}}} &= \frac{1}{S_y\sqrt{1 - \left(\frac{\Cov(Y, X)}{S_xS_y}\right)^2}}\\
&=\frac{1}{S_y\frac{\sqrt{S_x^2S_y^2 -\Cov^2(Y, X)}}{S_xS_y}}\\
&=\frac{1}{\frac{\sqrt{S_x^2S_y^2 -\Cov^2(Y, X)}}{S_x}}\label{c}\tag{3}
\end{align}
I tried to expand the $\Cov(Y, X)$ term and left $SSE$ and $SSX$ terms in $\ref{c}$, but there is no any further process.
I am wondering how to continue from $\ref{c}$, or my initial direction of the proof is not correct?
 [/INST] The trick is the following equation:$$\sum(\hat{Y_i} - Y_i)^2 = SSE = (1 - R^2)SST$$
This is why there is a $\sqrt{1 - \Cor^2(Y,X)}$ term in $(2)$. The whole proof is below:
$$
\begin{align}
t &= \frac{\hat{\beta_1}}{\operatorname{se}(\beta_1)}\\
  &= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\sqrt{\frac{\sum(\hat{Y_i} - Y_i)^2}{n-2}\cdot\frac{1}{\sum(X_i - \overline{X})^2}}}\\
  &= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\sqrt{\frac{(1 - R^2)SST}{n-2}\cdot\frac{1}{\sum(X_i - \overline{X})^2}}}\\
  &= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\sqrt{\frac{(1 - R^2)S_y^2(n-1)}{n-2}\cdot\frac{1}{S_x^2(n-1)}}}\\
  &= \frac{\frac{\Cov(Y,X)}{S_x^2}}{\frac{S_y}{S_x}\frac{\sqrt{1 - R^2}}{\sqrt{n - 2}}}\\
  &= \frac{\frac{\Cov(Y,X)}{S_yS_x}\sqrt{n-2}}{\sqrt{1 - R^2}}\\
  &= \frac{\Cor(Y,X)\sqrt{n-2}}{\sqrt{1 - \Cor^2(Y,X)}}
\end{align}
$$
</s>