You didn't do the product rule correctly. You should have gotten F'(x)=sin(x)*-sin(x)+cos(x)*cos(x). Look at koot's post for clarification on how I got this.
The sin of pi/3=sqrt(3)/2, and the cos of pi/3=1/2. It's just something that you memorize.
Take a look at this for help:
Your answer should be -1/2