https://file.aiquickdraw.com/mathbot/file/1729095699532wkvlx9ge.jpg

The image presents a mathematical proof that aims to demonstrate the identity:

\[
\cosh(x) + \sinh(x) = e^x
\]

**Step 1**: Using the definitions of the hyperbolic functions:
- \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)
- \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)

Substituting these into the equation:

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

**Step 2**: Combine the terms inside the parentheses:

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

**Step 3**: Simplify the expression:

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

**Step 4**: Simplify further:

\[
\cosh(x) + \sinh(x) = e^x
\]

This completes the proof.

Would you like more details or have any questions on this? Here are five related questions to consider:

1. What are the definitions of other hyperbolic functions like \(\tanh(x)\) and \(\coth(x)\)?
2. How are hyperbolic functions related to trigonometric functions?
3. Can you derive the identity \(\cosh^2(x) - \sinh^2(x) = 1\)?
4. What is the geometric interpretation of hyperbolic functions?
5. Can you prove other similar identities involving hyperbolic functions?

**Tip**: Remember, hyperbolic functions often arise in solutions to differential equations and in the study of special curves like catenaries.

Prove the identity: cosh(x) + sinh(x) = e^x

Step-by-step proof of the identity cosh(x) + sinh(x) = e^x using the definitions of hyperbolic functions. This problem highlights key concepts in hyperbolic and exponential functions, suitable for advanced high school or college-level students.

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