A wave represented by the equation y_1=a (k x-ω t) is superimposed...

Since the point $x=0$ is a node and reflection is taking place from point $x=0$. This means that reflection must be taking place from the fixed end and hence the reflected ray must suffer an additional phase change of $\pi$ or a path change of $\frac{\lambda}{2}$. So, if $y_{\text{incident }}=a \cos (k x-\omega t)$$\begin{aligned} \Rightarrowy_{\text{incident }} & =a \cos(-k x-\omegat+\pi)\\ & =-a \cos (\omegat+k x) \end{aligned}$$Hence equation$ for $the other wave y=a \cos (k x+$\omega$t+$\pi)