Suppose we have performed a simple linear regression with the least square method and have deduced the desired parameters to set up the two regression equations. $r_{xy}$ is the correlation coefficient, $r_{xy}:= \frac{s_{xy}}{s_x s_y}$ where $s_{xy}$ is the covariance. $s_x$ and $s_y$ are the standard deviations of all $x_i$ and $y_i$ samples and $\bar{x}$ and $\bar{y}$ are the means.
We must write the angle between the two regression lines as a function of $s_{xy}, s_x, s_y$. We are given the hint that it might be helpful to think of the tangent addition formula.
My approach:
The regression with dependent variable $y$ is:
$y = r_{xy}\frac{s_y}{s_x} x + \bar{y}-r_{xy}\frac{s_y}{s_x}\bar{x}$.
The regression with dependent variable $x$:
$x = r_{xy}\frac{s_x}{s_y} y + \bar{x}-r_{xy}\frac{s_x}{s_y}\bar{y}$.
I know that the angle $\varphi$ can be calculated via: $\varphi=\arccos\left(\frac{\langle u,v\rangle}{\Vert u\Vert \Vert v \Vert}\right)$. If I simply plug in the vectors: $u = {1 \choose r_{xy}\frac{s_y}{s_x}}$ and $ v = {1 \choose r_{xy}\frac{s_x}{s_y}}$ I get a messy term and I am not sure if this is the right way I should go!?
Have you any tips? Or is it the wrong way I am heading for?
PS: This is homework so please do not post the full solution unless it is hided.