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Hyperbolic cosine
The hyperbolic cosine can be expressed with exponential functions as
Explanation
In the power series for the hyperbolic cosine are only even exponents and all terms have a plus sign
After duplicating all terms you get
To this we add odd exponents and subtract these immediately
Rearranging gives
In braces are two power series for exponential functions which means
so that
Example 1
You can see that cosh (0) = 1, because