How to derive the formulas for hyperbolic cosine?

The formula for hyperbolic cosine can be derived using the exponential function.

To derive the formula for hyperbolic cosine, we start with the definition of the exponential function:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

Next, we substitute -x for x to get:

e^-x = 1 - x + (x^2)/2! - (x^3)/3! + ...

We then add these two equations together to get:

e^x + e^-x = 2 + (x^2)/2! + (x^4)/4! + ...

We can rearrange this equation to get:

(e^x + e^-x)/2 = 1 + (x^2)/2! + (x^4)/4! + ...

Now, we define the hyperbolic cosine function as:

cosh(x) = (e^x + e^-x)/2

Substituting this definition into the equation above, we get:

cosh(x) = 1 + (x^2)/2! + (x^4)/4! + ...

This is the formula for hyperbolic cosine. We can also derive the formula using the identity:

cosh(x) = cos(ix)

where i is the imaginary unit. Using the formula for cosine, we get:

cosh(x) = cos(ix) = (e^(ix) + e^(-ix))/2

Substituting x = iy, where y is a real number, we get:

cosh(iy) = (e^(iy) + e^(-iy))/2

Using Euler's formula, e^(iy) = cos(y) + i sin(y), we get:

cosh(iy) = (cos(y) + i sin(y) + cos(y) - i sin(y))/2

Simplifying this expression, we get:

cosh(iy) = cos(y)

Substituting x = iy back into the original formula, we get:

cosh(x) = cosh(iy) = cos(y)

Using the identity cosh^2(x) - sinh^2(x) = 1, we can derive the formula for hyperbolic sine:

sinh(x) = sqrt(cosh^2(x) - 1)

This completes the derivation of the formulas for hyperbolic cosine and sine.