Answer
Graph(III)
Work Step by Step
Write the parametric equations of a circle whose radius is defined as $r$ as follows: $x=r \cos t ; y =r \sin t$
We are given that $x= \cos^2 t , y=\sin^2 t, z=t$
Here both $x$ and $y$ are always positive on the part of the plane $x+y=1$ and vary from $0$ to $1$.
Also, as $t$ increases, the value of $x$ goes from $1$ to $0$ and the value of $y$ goes from $0$ to $1$.
In a similar manner, if the value of $t$ changes (or, decreases) in the other direction, then the value of $x$ goes from $0$ to $1$, while $y$ goes from $1$ to $0$.
Thus we have a sine wave, which matches with $Graph(III)$.
