Correlation coefficient is the cos between two series if they are treated as vectors (with nth data point being nth dimension of a vector). The above formula simply creates a decomposition of a vector into its $\cos\theta$, $sin\theta$ components (with respect to $X_1,X_2$).
if $\rho = cos \theta$ , then $\sqrt{1-{\rho}^2}=\pm sin \theta$.

Because if $X_1, X_2$ are uncorrelated, the angle between them is a right angle (ie, they can be considered as orthogonal, albeit non-normalized, basis vectors ).

Correlation coefficient is the $\cos$ between two series if they are treated as vectors (with $n^{th}$ data point being $n^{th}$ dimension of a vector). The above formula simply creates a decomposition of a vector into its $\cos\theta$, $sin\theta$ components (with respect to $X_1,X_2$).
if $\rho = cos \theta$ , then $\sqrt{1-{\rho}^2}=\pm sin \theta$.

Because if $X_1, X_2$ are uncorrelated, the angle between them is a right angle (ie, they can be considered as orthogonal, albeit non-normalized, basis vectors ).

Correlation coefficient is the cos between two series if they are treated as vectors (with nth data point being nth dimension of a vector). The above formula simply creates a decomposition of a vector into its cos\theta, sin\theta compoments (with respect to another vector).
if \rho = cos \theta , then \sqrt{1-{\rho}^2}=\pm sin \theta.

Because if X_1, X_2 are uncorrelated, the angle between them is a right angle (ie, the corr coef = 0).

Correlation coefficient is the cos between two series if they are treated as vectors (with nth data point being nth dimension of a vector). The above formula simply creates a decomposition of a vector into its $\cos\theta$, $sin\theta$ components (with respect to $X_1,X_2$).
if $\rho = cos \theta$ , then $\sqrt{1-{\rho}^2}=\pm sin \theta$.

Because if $X_1, X_2$ are uncorrelated, the angle between them is a right angle (ie, they can be considered as orthogonal, albeit non-normalized, basis vectors ).