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The question in the image is asking to find the value of \(\cos^2 \theta + \sin^2 \theta\) for a double angle situation where \(\theta = A + B\).

Let's analyze and solve it.

The given expression is:

\[
\frac{\sin A + \sin B}{\cos A + \cos B} = \tan \left(\frac{A + B}{2}\right)
\]

We are to find the value of \(\cos^2 \theta + \sin^2 \theta\).

### Solution:
1. The identity \(\cos^2 \theta + \sin^2 \theta = 1\) holds true for any angle \(\theta\).

2. Thus, without needing to delve deeper into the trigonometric manipulations for this specific problem, the answer is:
   \[
   \cos^2 \theta + \sin^2 \theta = 1
   \]

This identity is fundamental and applies universally.

### Summary:
The value of \(\cos^2 \theta + \sin^2 \theta\) is \(1\).

Would you like a more detailed explanation or have any questions? Here are some related questions:

1. What is the double angle identity for sine and cosine?
2. How do we derive the identity \(\cos^2 \theta + \sin^2 \theta = 1\)?
3. Can you explain the relationship between \(\tan \theta\) and \(\sin \theta\) and \(\cos \theta\)?
4. What are some applications of trigonometric identities in solving equations?
5. How does the identity \(\cos^2 \theta + \sin^2 \theta = 1\) apply to different angles?

**Tip:** Remember, the identity \(\cos^2 \theta + \sin^2 \theta = 1\) is one of the Pythagorean identities and is very useful in simplifying trigonometric expressions.

Explore how to find the value of cos² θ + sin² θ using the fundamental trigonometric identity. This problem involves understanding trigonometric functions and their basic identities. Suitable for Grades 10-12 students.

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