traveling waves

SInce you did not give the wave function y1(x,t) and y2(x,t), which are said in your question be given in Parts A and B respectively, I assume that they are the standard wave equation on a string. That is,

y1(x,t) = A.sin(kx - wt) and y2(x,t) = A.sin(kx + wt)
The equations represent two identical transverse waves travelling in opposite directions.

The sum of these two waves ys(x,t) is thus,
ys(x,t) = A.sin(kx - wt) + A.sin(kx + wt)
ys(x,t) = A[sin(kx).cos(wt) - cos(kx).sin(wt) + sin(kx).cos(wt) + cos(kx.)sin(wt)]
hence, ys(x,t) = A[2.sin(kx)cos(wt)
ys(x,t) = [2A.sin(kx)].cos(wt)

Compare with ys(x,t) = ye(x).yt(t)
Therefore, ye(x) = 2A.sin(kx) and yt(t) = cos(wt)