Hyperbolic thinking

Explore the properties of these two fascinating functions using trigonometry as a guide.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

This problem naturally follows on from Trig Reps, although the two problems may be attempted independently.

 

Steve left the following cryptic page in his notebook:

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Hyperbolic thinking


It seems that Steve thinks the following functions $A(x)$ and $B(x)$ are similar in some way to $\sin(x)$ and $\cos(x)$:

$$A(x) = \frac{1}{2}\Big(10^{x} +10^{-x}\Big)\quad\quad B(x) = \frac{1}{2}\Big(10^{x} -10^{-x}\Big)$$

Is Steve correct? To answer this question, think of as many properties of $\sin(x)$ and $\cos(x)$ as you can and, using these as a guide, explore the properties of $A(x)$ and $B(x)$.

 

Once you have done this you might wish to consider the properties of functions similar to $A(x)$ and $B(x)$ where the $10$ is replaced by different numbers. Do any of the properties hold for all of the bases? Which properties are base dependent? Is there a natural choice of base which the structure reveals?