Math Problem Statement
Solution
To prove the identity for hyperbolic functions:
we can use the definitions of the hyperbolic sine and cosine functions:
Proof:
Start with the left-hand side:
Now expand this expression:
Now, let's express the right-hand side of the identity we want to prove:
Using the definitions of and , we substitute:
and
Now, add the two expressions:
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
Related Recommendation
Proof of the Hyperbolic Identity sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)
Proving cosh(x) + sinh(x) = e^x Using Hyperbolic Function Identities
Proving Hyperbolic Double Angle Identities: sinh(2x) and cosh(2x)
Simplifying sinh(5x)cosh(2x) Using Hyperbolic Identities
Integration of sinh(x) cosh(x) with respect to x