If we for the moment ignore the domain interval, we know that cosine ranges between -1 and 1 so the range on this basis would be [-¾,¾] because of the coefficient. cos(2x-π/3)=-1 when 2x-π/3=(2n+1)π where n is an integer, so 2x=π/3+(2n+1)π, x=π/6+(2n+1)π/2; and cos(2x-π/3)=1 when 2x-π/3=2nπ, x=π/6+nπ. Now look at the given domain. Assuming -π2 means -π/2, we can see that if n=-1, x=π/6-π/2, that is, x>-π/2 so it's above the low side of the interval and so lies within the domain. That means the low end of the range of the given function is -¾.
When n=0, x=π/6 or ⅔π, therefore x<π and is within the given domain. So we've shown that the upper side of the range is ¾. A sketched graph will confirm that there are three values of x (-⅓π, ⅙π, ⅔π) within the domain that allow the full range to be [-¾,¾] (a total extent of 3/2 units).