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?
It is probably better to think of the difference in algebraic sign as what distinguishes trigonometry from hyperbolic trigonometry. Consider what happens when $\theta$ is close to zero. What happens to $\sin\theta$ and $\sinh\theta$? Same question for $\cos\theta$ and $\cosh\theta$. Those are the properties that you should consider the most basic.
If you understand complex numbers and the natural logarithm, it will be helpful to look at Euler's formula https://brilliant.org/wiki/hyperbolic-trigonometric-functions/ Notice the expressions for $\cosh$ and $\sinh$ are simpler than those for $\cos$ and $\sin$.