Question

Which of the six trigonometric functions do not depend on the value of r?

Answer 1

If we construct a triangle in a circle of radius $r$, the radius of the circle will be the hypotenuse of our triangle, and the legs will be $x$ and $y$ as you can see in the picture. 
Now we can construct our trigonometric ratios for our triangle as usual:
$sin \alpha = \frac{y}{r}$
$cos \beta = \frac{x}{r}$
$tan \beta = \frac{y}{x}$
$csc \beta = \frac{r}{y}$
$sec \beta = \frac{r}{x}$
$cot \beta = \frac{x}{y}$

As you can see the only two trigonometric functions that don't depend on the value of $r$ are tangent and cotangent.