Since x=cos^{2}(t) and y=sin^{2}(t), it follows that x+y=cos^{2}(t)+sin^{2}(t)=1 for all values of t. Therefore, the motion is always on the straight line with xy-equation x+y=1, which is equivalent to y=-x+1 (a straight line with a slope of -1 and a y-intercept of 1).
Also, since cos^{2}(t)\geq 0 and sin^{2}(t)\geq 0 for all t, this motion is always in the 1st quadrant of the plane where x\geq 0 and y\geq 0.
Now think about, for 0\leq t\leq 2pi, how the values of cos^{2}(t) oscillate from 1 to 0 to 1 to 0 and back to 1 again, while the values of sin^{2}(t) oscillate from 0 to 1 to 0 to 1 and back to 0 again. In other words, the motion traverses the line segment from (1,0) to (0,1) four times. By the Pythagorean Theorem (draw an appropriate right triangle), this line segment has length sqrt(1^{2}+1^{2})=sqrt(2).
This leads us to conclude that the total distance traveled (arc length) is 4sqrt(2).