The way to do this is to find out what two angles add up to 105. We need angles we already know the sine or cosine of. Let's list the most popular: 30, 60, 45, 90, 120, and so on. As it happens 60+45=105, so cos(105)=cos(60+45).
We can expand this: cos60co45-sin60sin45. Sin45=cos45=√2/2 and cos60=½ and sin60=√3/2.
Cos105=√2/4-√2√3/4=(√2/4)(1-√3). This is an exact value without calculating √2 or √3. This results in a negative value because √3>1. The equation can also be written cos105=(√2-√6)/4.