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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.

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