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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?

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    $\begingroup$ Seems fine, probably the fastest way too. $\endgroup$
    – vitamin d
    Commented Mar 24, 2021 at 23:13
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    $\begingroup$ Use the relationship $\cos x=\cosh ix, \sin x=-i\sinh ix$ and then you can convert any trig identity into a hyperbolic one. $\endgroup$
    – Paramanand Singh
    Commented Mar 25, 2021 at 3:15
  • $\begingroup$ You might consider Osborn's Rule [of Thumb] (see this answer). For a visual mnemonic, see the edit to this answer, which depicts the hyperbolic trig functions as segments related to a hyperbola; it's a counterpart to what I call the "Fundamental Trigonograph" for circular trig functions (as mentioned, eg, in this answer, which provides a link to a note with an in-depth discussion). $\endgroup$
    – Blue
    Commented Mar 25, 2021 at 4:43

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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 $\sin$.

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