How to derive the formulas for hyperbolic tangent?

The formula for hyperbolic tangent can be derived from the exponential function.

The hyperbolic tangent function is defined as:

tanh(x) = (e^x - e^-x) / (e^x + e^-x)

To derive this formula, we start with the exponential function:

e^x = lim(n->∞) (1 + x/n)^n

We can rewrite this as:

e^x = lim(n->∞) (1 + (2x/n))/(1 - (2x/n))^n/2

Using the binomial theorem, we can expand the numerator and denominator:

e^x = lim(n->∞) (1 + (2x/n) + (2x/n)^2/2! + ...)/(1 - (2x/n) + (2x/n)^2/2! - ...)^n/2

Taking the limit as n approaches infinity, we get:

e^x = (1 + x + x^2/2! + ...) / (1 - x + x^2/2! - ...)

Now, we can use this to derive the formula for hyperbolic tangent:

tanh(x) = (e^x - e^-x) / (e^x + e^-x)

tanh(x) = [(1 + x + x^2/2! + ...) - (1 - x + x^2/2! - ...)] / [(1 + x + x^2/2! + ...) + (1 - x + x^2/2! - ...)]

tanh(x) = (2x + 2x^3/3! + ...) / (2 + 2x^2/2! + ...)

tanh(x) = x + x^3/3 + x^5/5 + ...

Therefore, the formula for hyperbolic tangent is:

tanh(x) = x + x^3/3 + x^5/5 + ...