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please answer that provide a step by step solution 

Let's break down each part of the problem step-by-step.

### a) Prove that:
\[
-2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \sin(-x)
\]

#### Step 1: Use the double angle identity
We recognize that the left-hand side can be simplified using the double-angle identity for sine:
\[
\sin(2\theta) = 2 \sin(\theta) \cos(\theta)
\]
So, we set \(\theta = \frac{x}{2}\), and then:
\[
2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \sin(x)
\]
Thus:
\[
-2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = -\sin(x)
\]

#### Step 2: Use the odd property of the sine function
We know that sine is an odd function, meaning \(\sin(-x) = -\sin(x)\). Therefore:
\[
-\sin(x) = \sin(-x)
\]

Thus, the left-hand side simplifies to the right-hand side:
\[
-2 \sin\left(\frac{x}{2}\right) \cos\left(\frac{x}{2}\right) = \sin(-x)
\]

Hence, part (a) is proven.

---

### b) Prove that:
\[
\sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) = 1
\]

#### Step 1: Use the Pythagorean identity
One of the most fundamental trigonometric identities is the Pythagorean identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
This identity holds for any angle \(\theta\). So, if we let \(\theta = \frac{x}{2}\), we directly get:
\[
\sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) = 1
\]

Hence, part (b) is proven.

---

### Summary:
- For part (a), we used the double-angle identity for sine and the fact that sine is an odd function.
- For part (b), we directly applied the Pythagorean identity.

Do you have any further questions or need clarification on any part?

### Related Questions:
1. Can you derive the double-angle identity for sine from first principles?
2. What are the half-angle formulas for sine and cosine, and how are they derived?
3. Can you explain why sine is an odd function and cosine is an even function?
4. How would you solve similar problems involving double or half-angle identities for tangent?
5. What are the practical applications of using these trigonometric identities in physics or engineering?

### Tip:
Whenever you are unsure how to simplify a trigonometric expression, look for opportunities to apply standard identities like the Pythagorean, double-angle, or half-angle formulas.

Using half-angle formulas, show that a) -2sin(x/2)cos(x/2) = sin(-x), b) sin^2(x/2) + cos^2(x/2) = 1

Learn how to use half-angle formulas and trigonometric identities to prove expressions such as -2sin(x/2)cos(x/2) = sin(-x) and sin^2(x/2) + cos^2(x/2) = 1. This problem demonstrates key concepts in trigonometry, including double-angle identities and the Pythagorean identity.

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