Math Problem Statement

Prove that cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y).

Solution

To prove the identity for hyperbolic functions:

cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y),\cosh(x + y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y),

we can use the definitions of the hyperbolic sine and cosine functions:

cosh(z)=ez+ez2,sinh(z)=ezez2.\cosh(z) = \frac{e^z + e^{-z}}{2}, \quad \sinh(z) = \frac{e^z - e^{-z}}{2}.

Proof:

Start with the left-hand side:

cosh(x+y)=ex+y+e(x+y)2.\cosh(x + y) = \frac{e^{x+y} + e^{-(x+y)}}{2}.

Now expand this expression:

cosh(x+y)=exey+exey2.\cosh(x + y) = \frac{e^x e^y + e^{-x} e^{-y}}{2}.

Now, let's express the right-hand side of the identity we want to prove:

cosh(x)cosh(y)+sinh(x)sinh(y).\cosh(x) \cosh(y) + \sinh(x) \sinh(y).

Using the definitions of cosh\cosh and sinh\sinh, we substitute:

cosh(x)cosh(y)=ex+ex2ey+ey2=(exey+exey+exey+exey)4,\cosh(x) \cosh(y) = \frac{e^x + e^{-x}}{2} \cdot \frac{e^y + e^{-y}}{2} = \frac{(e^x e^y + e^x e^{-y} + e^{-x} e^y + e^{-x} e^{-y})}{4},

and

sinh(x)sinh(y)=exex2eyey2=(exeyexeyexey+exey)4.\sinh(x) \sinh(y) = \frac{e^x - e^{-x}}{2} \cdot \frac{e^y - e^{-y}}{2} = \frac{(e^x e^y - e^x e^{-y} - e^{-x} e^y + e^{-x} e^{-y})}{4}.

Now, add the two expressions:

cosh(x)cosh(y)+sinh(x)sinh(y)=exey+exey+exey+exey4+exeyexeyexey+exey4.\cosh(x) \cosh(y) + \sinh(x) \sinh(y) = \frac{e^x e^y + e^x e^{-y} + e^{-x} e^y + e^{-x} e^{-y}}{4} + \frac{e^x e^y - e^x e^{-y} - e^{-x} e^y + e^{-x} e^{-y}}{4}.

Combining terms:

= \frac{2(e^x e^y + e^{-x} e^{-y})}{4} = \frac{e^x e^y + e^{-x} e^{-y}}{2}.$$ This is exactly the same expression as the left-hand side $$\cosh(x + y)$$. Thus, we have shown that: $$\cosh(x + y) = \cosh(x) \cosh(y) + \sinh(x) \sinh(y).$$ Would you like any further explanation or clarification? Here are some related questions for deeper exploration: 1. What are the definitions of hyperbolic sine and cosine functions? 2. How does the hyperbolic cosine function relate to exponential functions? 3. Can you prove a similar identity for $$\sinh(x + y)$$? 4. How do hyperbolic identities compare to trigonometric identities? 5. What are the applications of hyperbolic functions in physics or engineering? **Tip:** The hyperbolic identities are closely analogous to the trigonometric addition formulas, with $$\cosh(x)$$ and $$\sinh(x)$$ playing similar roles to $$\cos(x)$$ and $$\sin(x)$$, respectively.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Hyperbolic Functions
Algebra

Formulas

cosh(x) = (e^x + e^(-x)) / 2
sinh(x) = (e^x - e^(-x)) / 2

Theorems

Addition formulas for hyperbolic functions
Hyperbolic identities

Suitable Grade Level

Grades 11-12, University Level