Answer
The wave equation is satisfied.
Work Step by Step
We need to prove the wave equation $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
We take the first partial derivative of the given function $w(t,x)$ with respect to $x$, by treating $t$ as a constant, and vice versa:
$\dfrac{\partial w}{\partial t}=c \cos (x+ct)$ and $\dfrac{\partial w}{\partial x}=\cos (x+ct)$
Now, $c^2 \dfrac{\partial^2 w}{\partial x^2}=c^2 \dfrac{\partial}{\partial x}(\dfrac{\partial w}{\partial x})=c^2 [-\sin (x+ct)]=-c^2 \sin (x+ct)$
So, $c^2 \dfrac{\partial^2 w}{\partial x^2}= \dfrac{\partial^2 w}{\partial t^2}$
Thus, it has been proved that the wave equation is satisfied.