$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$

$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$

I was playing around with trigonometric functions when I stumbled across
this $$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$$ Immediately I
checked it to see if it was flawed so I devised a proof $$\begin{align}
\sec^2\theta+\csc^2\theta&=\sec^2\theta\csc^2\theta\\
\frac1{\cos^2\theta}+\frac1{\sin^2\theta}&=\frac1{\cos^2\theta\sin^2\theta}\\
\frac{\cos^2\theta+\sin^2\theta}{\cos^2\theta\sin^2\theta}&=\frac1{\cos^2\theta\sin^2\theta}\\
\frac1{\cos^2\theta\sin^2\theta}&=\frac1{\cos^2\theta\sin^2\theta}\blacksquare\\
\end{align}$$ I checked over my proof many times and I couldn't find a
mistake, so I assume that my claim must be true.
So my questions are:
Is there a deeper explanation into why adding the squares is the same as
multiplying?
Is this just a property of these trigonometric functions or do similar
relationships occur with other trigonometric functions?
And finally as an additional curiosity what does this translate into
geometrically?