# Relationship between Covariance, Variance and Correlation

I am studying these slides : Random Forest. Slide 6 explains how to reduce the variance and at some point in the mathematical derivation, the covariance equals the variance times correlation (line 3) can someone explain this step mathematically?

In general, if two variables $$X,\,Y$$ have standard deviations $$\sigma_X,\,\sigma_Y$$ and correlation coefficient $$\rho$$, their covariance is $$\rho\sigma_X\sigma_Y$$. Each $$T_i$$ has variance $$\sigma^2$$ and standard deviation $$\sigma>0$$, so if $$T_i,\,T_j$$ have correlation $$\rho$$ their covariance is $$\rho\cdot\sigma\cdot\sigma=\sigma^2\rho$$.
In general if $$X,Y$$ are non-degenerate random variables defined on the same probability space that both have a second moment then the correlation of $$X$$ and $$Y$$ is defined by:$$\rho(X,Y)=\frac{\mathsf{Cov(X,Y)}}{\sigma_X\sigma_Y}$$where $$\sigma_X^2:=\mathsf{Var}(X)$$ and $$\sigma_Y^2:=\mathsf{Var}(Y)$$.
In special case $$\sigma_X=\sigma_Y=\sigma$$ this relation can be written as:$$\sigma^2\rho(X,Y)=\mathsf{Cov}(X,Y)$$