Hmm, interesting! I'm sure there are many ways to prove that ABC is a right triangle, but here's one method:

So let's assume ABC is a right triangle.

is the hypotenuse of the triangle

and are both legs of the triangle

Leg is the opposite side to angle ; leg isn't adjacent to angle . So the sine of A would be .

can be expanded to

The cosine of is only if angle is or Since a triangle cannot have a angle, angle must be , and ABC is a right triangle.