You can put this solution on YOUR website! Whenever a Trig. problem refers to "exact value" it is telling you that that the value can be found without calculators. And this means that "special" triangles are involved.
Let's look at a diagram of your problem. When using diagrams it is most helpful to include the unit circle (a circle of radius 1) because it makes the Trig. ratios easier to find.
For any point on the unit circle, the x-coordinate is the cos and the y-coordinate is the sin of the angle whose side passes through that point. The side of our 300 degree angle intersects the unit circle at point A. So the x-coordinate of point A is the cos(300). Now we just need to find the x-coordinate of point A!
Since our angle is 300 degrees, the angle AOB must be 60 degrees. Angle OBA is a right angle so angle OAB must be 30 degrees. That makes triangle AOB a 30-60-90 right triangle. From Geometry we know that there are certain relationships between the sides of 30-60-90 right triangles. One relationship is that the hypotenuse (OA) is always twice as long as the side opposite the 30 degree angle (OB). Putting this the other way around, the side opposite the 30 degree angle is one half as long as the hypotenuse. And since we are using a unit circle, our hypotenuse is 1. So that make OB 1/2. And OB is the x-coordinate of A so cos(300) is 1/2.
(