Ok, to get the answer without using a calculator, you first need to draw a right angled triangle with a 30 degree angle. Using a bit of logic and tan(30)=O/A, you will figure out that tan(30)=√3/3. tan(45)=1, this is quite famous.
You can get the answer to tan(15) by using tan(45-30) and the trig identity (tanA-tanB)/(1+tanAtanB).
From there you can produce the result 2-√3=tan(45-30)=tan(15)=tan(π/12)
