I'm working on the following problem:
I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the right direction. Here's my work so far:
\begin{align} \cos(z)&=\cos(x)\cosh(y)-i \sin(x)\sinh(y)\\ \text{sec}(z)=\frac{1}{\cos(x)} &=\frac{1}{\cos(x)\cosh(y)-i \sin(x)\sinh(y)} \cdot \frac{\cos(x)\cosh(y)+i \sin(x)\sinh(y)}{\cos(x)\cosh(y)+i \sin(x)\sinh(y)}\\ &=\frac{\cos(x)\cosh(y)+i \sin(x)\sinh(y)}{\cos(x)^2\cosh^2(y)+\sin(x)^2\sinh^2(y)}\\ \end{align}
Thus, I need only show that $\cos(x)\cosh(y) - i\sin(x)\sinh(y) = -1+\sin^2(x)-\sinh^2(y)$. However, this is not true, per my previous post.