= Solution
The two terms in the preceding formula are the <real part> and <imaginary part>. Their squared sum is the squared <complex modulus>:
$$
\begin{aligned}
|\sinh(x+iy)|^2&=\sinh^2x\cos^2y+\cosh^2x\sin^2y\\
&=\sinh^2x(\cos^2y+\sin^2y)+\sin^2y\\
&=\sinh^2x+\sin^2y.
\end{aligned}
$$
Here $\cosh^2x-\sinh^2x=1$ follows directly from the <exponential function> definitions. The <hyperbolic sine modulus identity> therefore yields
$$
\boxed{|\sinh(x+iy)|=\sqrt{\sinh^2x+\sin^2y}\geq|\sinh x|.}
$$
Equality holds exactly when $\sin y=0$, that is, when $y$ is an integer multiple of $\pi$. The lower bound is uniform in $y$, which will control whole vertical sides of the <contour> rather than just their endpoints.
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