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The image you've uploaded contains mathematical expressions. Here's a breakdown of what I can see:

1. \( f\left(\frac{\pi}{2}\right) + f\left(\sin \frac{\pi}{2}\right) = \cos x + 1 \)
   
2. You are asked to find \( f(x) \) and \( f\left(\cos \frac{x}{2}\right) \).

This looks like a functional equation problem where we need to solve for the function \( f(x) \) based on the given equation.

Would you like a detailed solution for this, or should I proceed step-by-step? 

Additionally, here are 5 related questions that could deepen your understanding of functional equations:

1. How do we approach solving a functional equation?
2. What properties of trigonometric functions (like sine and cosine) are useful in these types of problems?
3. Could there be multiple functions \( f(x) \) that satisfy this equation?
4. What role does substitution play in solving functional equations?
5. How can we verify the solution once we find \( f(x) \)?

**Tip:** In functional equations, always test simple values like \( 0 \), \( 1 \), or known constants (like \( \pi \) or \( \frac{\pi}{2} \)) to find specific information about the function!

Find the function \( f(x) \) given the equation: \( f\left(\frac{\pi}{2}\right) + f\left(\sin \frac{\pi}{2}\right) = \cos x + 1 \). Also, find \( f(\cos \frac{x}{2}) \).

Explore how to solve a functional equation involving trigonometric functions, such as sine and cosine. The problem requires finding the function f(x) given a functional relationship and also determining f(cos(x/2)). Suitable for undergraduate-level students.

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