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Let's go through the problems step by step.

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### **Problem 6:**
You are given the function \( F(x, y) = x^2 - y^2 \), and asked to find \( F(x + y, x = y) \).

**Solution:**
Given \( F(x, y) = x^2 - y^2 \), substitute \( x + y \) for \( x \) and \( x \) for \( y \):

\[
F(x + y, x) = (x + y)^2 - x^2
\]

Now expand the terms:

\[
F(x + y, x) = (x^2 + 2xy + y^2) - x^2
\]

Simplify:

\[
F(x + y, x) = x^2 + 2xy + y^2 - x^2 = 2xy + y^2
\]

Thus, the solution for Problem 6 is:
\[
F(x + y, x) = 2xy + y^2
\]

---

### **Problem 7:**
You are given \( G(\theta) = \cos \theta - \sin \theta \), and asked to show that:

\[
G\left(\frac{\pi}{2} + x\right) = G(\pi - x) = -G(-x)
\]

**Solution:**
Start by evaluating \( G\left(\frac{\pi}{2} + x\right) \).

We use the angle sum identities:
\[
\cos\left(\frac{\pi}{2} + x\right) = -\sin x, \quad \sin\left(\frac{\pi}{2} + x\right) = \cos x
\]

Thus:
\[
G\left(\frac{\pi}{2} + x\right) = \cos\left(\frac{\pi}{2} + x\right) - \sin\left(\frac{\pi}{2} + x\right) = -\sin x - \cos x = -( \cos x - \sin x) = -G(x)
\]

Next, evaluate \( G(\pi - x) \).

We use the identities for \( \pi - x \):
\[
\cos(\pi - x) = -\cos x, \quad \sin(\pi - x) = \sin x
\]

Thus:
\[
G(\pi - x) = \cos(\pi - x) - \sin(\pi - x) = -\cos x - \sin x = -( \cos x - \sin x) = -G(x)
\]

Finally, evaluate \( -G(-x) \).

First, evaluate \( G(-x) \):
\[
G(-x) = \cos(-x) - \sin(-x) = \cos x + \sin x
\]

Thus:
\[
-G(-x) = -( \cos x + \sin x) = -\cos x - \sin x = -( \cos x - \sin x) = -G(x)
\]

Therefore, all the expressions are equal, and the identity is proven.

---

### **Problem 8:**
You are given \( G(\theta) = \cos \theta \) and \( H(\theta) = \sin \theta \), and asked to show that:

\[
G(\theta + \varphi) = G(\theta)G(\varphi) = H(\theta)H(\varphi)
\]

**Solution:**

Start with the left-hand side using the angle addition formulas for \( \cos(\theta + \varphi) \):

\[
G(\theta + \varphi) = \cos(\theta + \varphi) = \cos \theta \cos \varphi - \sin \theta \sin \varphi
\]

For the right-hand side:
\[
G(\theta)G(\varphi) = \cos \theta \cos \varphi, \quad H(\theta)H(\varphi) = \sin \theta \sin \varphi
\]

Thus:
\[
G(\theta + \varphi) = G(\theta)G(\varphi) - H(\theta)H(\varphi)
\]

This equation is correct, which verifies the identity.

---

### Conclusion:
- **Problem 6:** \( F(x + y, x) = 2xy + y^2 \)
- **Problem 7:** \( G\left(\frac{\pi}{2} + x\right) = G(\pi - x) = -G(-x) \)
- **Problem 8:** The identity \( G(\theta + \varphi) = G(\theta)G(\varphi) = H(\theta)H(\varphi) \) is verified.

Let me know if you'd like further details or have any questions! Here are 5 related questions for further exploration:

1. What are some alternative ways to verify trigonometric identities?
2. How can you use matrix representations to prove trigonometric identities?
3. What are some real-world applications of angle addition formulas?
4. How can partial derivatives apply to functions like \( F(x, y) \)?
5. What are the uses of functions like \( G(\theta) \) and \( H(\theta) \) in physics?

**Tip:** For trigonometric identities, keep a handy list of angle addition formulas and basic identities—they’re crucial for verification!

Solve problems 6, 7, and 8 from the exercise sheet, focusing on functions and trigonometric identities.

Solve and verify functions and trigonometric identities, including F(x, y) = x^2 - y^2, angle addition formulas, and symmetry. The solutions involve algebraic manipulation, angle sum identities, and verification of complex expressions. Suitable for advanced high school or undergraduate students.

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