Why is the cosine of a right angle, 90 degrees, equal to zero?

Others have mentioned that such a right angle triangle does not exist.

As a compromise how about

$$\cos(45+45)=\cos(45)\cos(45)-\sin(45)\sin(45)=0 $$

The addition formula is easily justified geometrically.

I chose $45^{\circ}$ because you can work this exactly with an isosceles right angled triangle of side length $1,1,\sqrt{2}$