101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y

Verified step by step guidance

1. Recall the definitions of hyperbolic cosine and hyperbolic sine: \(\cosh z = \frac{e^{z} + e^{-z}}{2}\) and \(\sinh z = \frac{e^{z} - e^{-z}}{2}\) for any variable \(z\).
2. Express \(\cosh(x + y)\) using its definition: \(\cosh(x + y) = \frac{e^{x+y} + e^{-(x+y)}}{2}\).
3. Rewrite the right-hand side of the identity using the definitions: \(\cosh x \cosh y + \sinh x \sinh y = \left(\frac{e^{x} + e^{-x}}{2}\right) \left(\frac{e^{y} + e^{-y}}{2}\right) + \left(\frac{e^{x} - e^{-x}}{2}\right) \left(\frac{e^{y} - e^{-y}}{2}\right)\).
4. Expand both products on the right-hand side carefully, combining like terms and simplifying the expression.
5. Show that after simplification, the right-hand side equals \(\frac{e^{x+y} + e^{-(x+y)}}{2}\), which matches the left-hand side, thus proving the identity.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh and cosh, are analogs of trigonometric functions but based on hyperbolas instead of circles. They are defined using exponential functions: cosh x = (e^x + e^{-x})/2 and sinh x = (e^x - e^{-x})/2. Understanding their definitions is essential for manipulating and proving identities involving them.

Addition Formulas for Hyperbolic Functions

Addition formulas express the value of hyperbolic functions at sums of arguments in terms of functions at individual arguments. For example, cosh(x + y) can be expanded using the definitions of cosh and sinh. These formulas are key tools for proving identities and simplifying expressions involving sums.

Algebraic Manipulation of Exponentials

Proving hyperbolic identities often requires rewriting functions in terms of exponentials and then applying algebraic operations like expansion, grouping, and factoring. Mastery of these algebraic techniques allows one to transform complex expressions into simpler, recognizable forms to verify identities.