Exercises
[1MD]We define the functions hyperbolic cosine 1
\[ \cosh y =\frac{e^{y}+e^{-y}} 2 \]and hyperbolic sine
\[ \sinh y =\frac{e^{y}-e^{-y}}{2}. \]Verify that
\[ (\cosh x) ^ 2 - (\sinh x) ^ 2 =1 \](which justifies the name of ”hyperbolic”).
Prove the validity of these power series expansion
\[ \cosh (x) = 1+\frac 1 2 x^ 2+ \frac 1{4!} x^ 4+\frac 1{6!} x^ 6+ \ldots \]\[ \sinh (x) = x+\frac 1{3!} x^ 3+ \frac 1{5!} x^ 5+\frac 1{7!} x^ 7+ \ldots \]Check that
\[ \cosh '=\sinh ~ ~ ,~ ~ \sinh '=\cosh \]Check the formulas
\[ \sinh (x+y) = \cosh x \sinh y + \cosh y \sinh x ~ ~ \]\[ \cosh (x+y) = \cosh x \cosh y + \sinh y \sinh x ~ . \]