I just tried to memorize hyperbolic formulas, and realized that there are lots of similarity between formulas for trigonometric and hyperbolic functions. for example: $$\cos^2x+\sin^2x=1\quad\quad\quad\quad\quad\quad\quad1+\tan^2x=\sec^2x$$ $$\cosh^2x-\sinh^2x=1\quad\quad\quad\quad\quad\quad\quad1-\tanh^2x=\text{sech}^2 x$$ And so on. When I look at it closer I see they are almost the same formulas but the sign comes before a $\sinh$ or $\tanh$ are different than the sign of $\sin$ ,$\tan$ in trig functions. To justify that, I considered the fact that parametric points $(\cos\theta,\sin\theta)$ placed on the circle $x^2+y^2=1$ and points $(\cosh \theta,\sinh \theta)$ are placed on hyperbola $x^2-y^2=1$, so we see the different signs comes before $y^2$ and because $\sin$ and $\sinh$ represents the vertical distance, these signs are different in all formulas.
Is my justification right?
Is there better way to justify this different in the formulas?