Hyperbolic square sum identity

\cosh^2x + \sinh^2x = \cosh(2x)

We know:
- \sinh x = \frac{e^x - e^{-x}}{2}
- \cosh x = \frac{e^x + e^{-x}}{2}
Square both functions:
- \sinh^2 x = \left(\frac{e^x - e^{-x}}{2}\right)^2 = \frac{e^{2x} - 2 + e^{-2x}}{4}
- \cosh^2 x = \left(\frac{e^x + e^{-x}}{2}\right)^2 = \frac{e^{2x} + 2 + e^{-2x}}{4}
\cosh^2 x + \sinh^2 x
- = \frac{e^{2x} + 2 + e^{-2x}}{4} + \frac{e^{2x} - 2 + e^{-2x}}{4}
- = \frac{e^{2x}+2+e^{-2x}+e^{2x}-2+e^{-2x}}4
- = \frac{2e^{2x}+2e^{-2x}}4
- = \frac{e^{2x}+e^{-2x}}2
- = \cosh(2x)
We’ve now shown thatt \cosh^2x + \sinh^2x = \cosh(2x)!

Question	Answer
What is the formula for \sinh x?	\sinh x = \frac{e^x - e^{-x}}{2}
What is the formula for \cosh x?	\cosh x = \frac{e^x + e^{-x}}{2}
What does \sinh^2 x equal when expanded using exponentials?	\sinh^2 x = \frac{e^{2x} - 2 + e^{-2x}}{4}
What does \cosh^2 x equal when expanded using exponentials?	\cosh^2 x = \frac{e^{2x} + 2 + e^{-2x}}{4}
What is the hyperbolic square sum identity?	\cosh^2x + \sinh^2x = \cosh(2x)
How do you prove \cosh^2x + \sinh^2x = \cosh(2x) using exponentials?	Sum the expanded squares: \frac{e^{2x} + 2 + e^{-2x}}{4} + \frac{e^{2x} - 2 + e^{-2x}}{4} = \frac{2e^{2x}+2e^{-2x}}4 = \frac{e^{2x}+e^{-2x}}2 = \cosh(2x)