Question 181075
In order to do this problems you must know the values of your special angles, otherwise you will keep having lots of trouble:

sin 0 = 0
sin30=1/2
sin45= sqr(2)/2
sin60 = sqr (3)/2
sin 90 = 1
cos 0 =1
cos30 = sqr(3)/2
cos45= sqr(2)/2
cos60 = 1/2
cos90= 0

Also you must know that pi = 180 degrees....so pi/3 = 180/3=60



.....NOW.....
sin^2 pi/3 + cos^2 pi/6 - sin^2 5pi/3

(sin pi/3)^2 + (cos pi/6)^2 - (sin 5pi/3)^2
(sin 180/3)^2 + (cos 180/6)^2 - (sin 5(180)/3)^2
(sin 60)^2 + (cos 30)^2 - (sin 300)^2
(sqr(3)/2)^2 + (sqr(3)/2)^2 - (-sqr(3)/2)^2
3/4 + 3/4 - 3/4
3/4

Note:  to find the value of sin 300 you must first find the reference angle of angle 300 which is 60.....and remember that angle 300 lies on the IV quadrant...
therefore sin300= -sin60