Apparent Contradiction in Inverting Linear Regression

Question

I believe this is a basic question, but I still can't seem to figure it out.

In linear regression, if the standard deviation of the x-data and the y-data is 1, the slope of the best fit line is the correlation of X, Y (because slope = cor(x, y) * (sd(x)/sd(y))). Therefore, because the correlation of two variables is between -1 and 1, the slope is between -1 and 1.

Now if we switch the x and y axes, the slope of the best fit line is the reciprocal of the previous one. Thus the new slope can be smaller than -1 and larger than 1. However, this is the same as finding the best fit line with the original X data on the Y axis and the Y data on the X axis. As before, this would give slope = cor(y, x) * (sd(y)/sd(x)). Since the standard deviations are still 1, slope = cor(y, x) = cor(x, y). Thus the slope must be between -1 and 1. Where am I going wrong?

Answer 1

if we switch the x and y axes, the slope of the best fit line is the reciprocal of the previous one

No, it isn't, because the best fit regression line minimizes the sum of squares in the direction of the response variable. When you interchange the roles of $x$ and $y$ across predictor-variable and response-variable, the best fit regression line is no longer the same line:

So the slope does remain between -1 and 1.