Question

Prove cosh(2x) = cosh^(2)x + sin^(2)x

Answer 1

its just going back to the definitions

Answer 2

\[\cosh (nx) = \frac{1}{2} ( e^{nx} + e^{-nx} ) \]

Answer 3

\[\sinh(nx) = \frac{1}{2}(e^{nx} - e^{-nx} )\]

Answer 4

\[(\sinh(x))^2 + (\cosh(x))^2 = \frac{1}{4} ( e^{x} + e^{-x} ) ^2 + \frac{1}{4}(e^{x} - e^{-x})^2\]

Answer 5

then expand , you can do the rest , its pretty easy

Answer 6

all you need to know is (e^(x))^2 = e^(2x)

Answer 7

then alot of things will cancel and you will get \[\frac{1}{2}(e^{2x} +e^{-2x} ) \]

Answer 8

which is cosh(2x) by definition