Question

AB and AD are tangents of the circle with the center at C. The measure of BDC = 45o, and the circle has a diameter of 4. Which is the length of AB?

Answer 1

The Line from C perpendicular to DC will intersect DB (possibly extended) at a distance (radius) 2 from C because tan(45) = 1. But B is the only other point of DB at distance 2 from C, so BCD is a right angle.
That makes 3 of angles of ABCD right angles, so ABCD is a square and AB = 2.