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:
$\dfrac{\partial w}{\partial t}=-2c \sec^2 (2x-2ct) \\ \dfrac{\partial w}{\partial x}= 2 \sec^2 (2x-2ct)$
Now, $ \dfrac{\partial^2 w}{\partial t^2}= \dfrac{\partial}{\partial t}( -2c \sec^2 (2x-2ct))= 8c^2 \sec^2 (2x-2ct) \tan (2x-2ct)$
and $c^2 \dfrac{\partial^2 w}{\partial x^2}= 2c^2 \times [2 \sec (2x-2ct) sec (2x-2ct) tan (2x-2ct) (2)] = 8c^2 \sec^2 (2x-2ct) \tan (2x-2ct)$
Thus, it has been proved that the wave equation is satisfied.
