Question 600374
you say sin(5pie/3) cos(5pie/3)-cos(5pie/3) sin (pie/3), but i think the question was rather 
sin(5pie/3) cos(pie/3)-cos(5pie/3) sin (pie/3)

And if the question was what i just wrote , you do it this way.

first let take a look a those trigonometric formulas
 sin(a+b) = sin(a) cos(b) + sin(b) cos(a)
sin(a-b) = sin(a) cos(b) - sin(b) cos(a)
Cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
Cos(a-b) = cos(a) cos(b) + sin(a) sin(b)


Now, let's  get back to our problem 
 let's call 5pie/3  a and call pie/3 b
Then, sin5pie/3 cospie/3-cos5pie/3 sin pie/3= sin (a) cos(b)- cos(a) sin(b),  
This looks like the second formula up above
so we get sin5pie/3 cospie/3-cos5pie/3 sin pie/3 = Sin ( a-b)
 and we know for this case that a = 5pie\3 and b=pie/3
so, sin5pie/3 cospie/3-cos5pie/3 sin pie/3 = Sin ( 5pie/3 - pie/3)
                                           = sin ( 4pie/3)
                                           = -(sqrt(3))/2

So the anwer is Sin( 4pie/3) and if you want to go further and find its value it is -(sqrt(3))/2