https://r2.erweima.ai/mathbot/image/1726309147799mbwss8e6.jpg

The image contains a mathematical equation and some handwritten annotations. Let's analyze the given mathematical expressions:

1. **Main Equation:**
   \[
   f(x) = \cos x
   \]
   This function \( f(x) \) satisfies the following functional equation:
   \[
   f(x - y) = f(x) f(y) + g(x) g(y)
   \]
   where:
   \[
   f(0) = 1, \quad \lim_{x \to 0} g(x) = 1
   \]

2. **Handwritten Notes:**
   - It seems there is an equation where \( y = 1 \), resulting in:
     \[
     f(x + 1) = f(x) + a
     \]
   - Another part shows:
     \[
     f(2) - f(1) = a, \quad f(x) - f(x - 1) = a
     \]
   This suggests a recursive structure for the function \( f(x) \) with respect to a constant \( a \).

Would you like me to analyze these conditions further or break down any part in more detail? Here are five potential follow-up questions based on this:

1. How does the functional equation \( f(x - y) = f(x)f(y) + g(x)g(y) \) behave when substituting specific values for \( x \) and \( y \)?
2. What kind of functions satisfy the condition \( f(x + 1) = f(x) + a \)?
3. Can we solve for \( f(x) \) given the condition \( f(0) = 1 \)?
4. What additional constraints does \( \lim_{x \to 0} g(x) = 1 \) impose on the function \( g(x) \)?
5. How does the equation \( f(x) - f(x - 1) = a \) relate to the structure of the function \( f(x) \)?

### Tip:
Functional equations often lead to discovering the general form of a function, and identifying patterns such as recurrence relations can greatly simplify the process.

f(x) = cos(x) satisfies f(x - y) = f(x)f(y) + g(x)g(y), f(0) = 1, lim(x->0) g(x) = 1.

Learn how to solve the functional equation f(x) = cos(x) with the given conditions f(x - y) = f(x)f(y) + g(x)g(y), f(0) = 1, and lim(x->0) g(x) = 1. This solution explores trigonometric properties and recurrence relations in a step-by-step approach.

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