| sinh (z) |^2 = sinh^2(x) + sin^2(y)

**dwsmith** · December 21st 2010, 01:20 PM

I am having trouble proving this identity $|sinh(z)|^2=sinh^2(x)+sin^2(y) \ \ z\in\mathbb{C} \ \ x,y\in\mathbb{R}$ .

$\displaystyle \left(\sqrt{sinh^2(z)}\right)^2=\left(\frac{e^z-e^{-z}}{2}\right)^2=\frac{e^{2z}+e^{-2z}-2}{4}$

$\displaystyle =\frac{e^{2x}cos(2y)+e^{-2x}cos(2y)-2+\mathbf{i}(e^{2x}sin(2y)-e^{-2x}sin(2y))}{4}$

$\displaystyle =\frac{cos(2y)(e^{2x}+e^{-2x})-2+\mathbf{i}sin(2y)(e^{2x}-e^{-2x})}{4}$

$\displaystyle =\frac{cos(2y)cosh(2x)-2+\mathbf{i}sin(2y)sinh(2x)}{2}$

At this point, I am not sure how to proceed.

**Plato** · December 21st 2010, 02:16 PM

$\sinh(z)=\sinh(x)\cos(y)+i~\cosh(x)\sin(y)$ .
$|\sinh(z)|^2=\sinh^2(x)\cos^2(y)+\cosh^2(x)\sin^2( y)$
What can you do with that?

**dwsmith** · December 21st 2010, 02:23 PM

$sinh^2(x)cos^2(y)+cosh^2(x)sin^2(y)$

$cosh^2(x)=1+sinh^2(x), \ \ sinh^2(x)cos^2(y)+(1+sinh^2(x))sin^2(y)$

$=sinh^2(x)(cos^2(y)+sin^2(y))+sin^2(y)=sinh^2(x)+s in^2(y)$

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