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The image contains a mathematical problem asking to use **De Moivre's Theorem** together with the **Binomial Theorem** to derive trigonometric identities.

Let's break it down step by step.

### 1. **De Moivre's Theorem:**
De Moivre's Theorem states:
\[
(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)
\]
For \(n = 4\), we can expand \((\cos \theta + i \sin \theta)^4\).

### 2. **Using the Binomial Theorem:**
To expand \((\cos \theta + i \sin \theta)^4\), we use the binomial theorem:
\[
(\cos \theta + i \sin \theta)^4 = \sum_{k=0}^{4} \binom{4}{k} (\cos \theta)^{4-k} (i \sin \theta)^k
\]
This will give us terms involving powers of \(\cos \theta\), \(\sin \theta\), and powers of \(i\). The real part of this expansion will give \(\cos 4\theta\) and the imaginary part will give \(\sin 4\theta\).

### 3. **Deriving the Cosine and Sine Formulas:**

#### (i) Expansion of \( \cos 4\theta \):
We focus on the real terms of the expansion:
\[
\cos 4\theta = \binom{4}{0} \cos^4 \theta + \binom{4}{2} (-1) \cos^2 \theta \sin^2 \theta + \binom{4}{4} (-1)^2 \sin^4 \theta
\]
Simplifying each term:
\[
\cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta
\]
Thus, the identity for \(\cos 4\theta\) is:
\[
\cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta
\]

#### (ii) Expansion of \( \sin 4\theta \):
Now, focus on the imaginary part:
\[
\sin 4\theta = \binom{4}{1} 4 \cos^3 \theta \sin \theta + \binom{4}{3} (-1) 4 \cos \theta \sin^3 \theta
\]
Simplifying:
\[
\sin 4\theta = 4 \cos^3 \theta \sin \theta - 4 \cos \theta \sin^3 \theta
\]
Thus, the identity for \(\sin 4\theta\) is:
\[
\sin 4\theta = 4 \cos^3 \theta \sin \theta - 4 \cos \theta \sin^3 \theta
\]

### Final Answer:
Using **De Moivre's Theorem** and the **Binomial Theorem**, the required identities are derived:
\[
\cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta
\]
\[
\sin 4\theta = 4 \cos^3 \theta \sin \theta - 4 \cos \theta \sin^3 \theta
\]

Would you like more details or clarifications on any part?

Here are 5 related questions:
1. How does De Moivre's theorem apply to higher powers of complex numbers?
2. Can you generalize the formula for \(\cos(n\theta)\) and \(\sin(n\theta)\)?
3. How can we use the double-angle formulas to derive similar results?
4. What are the key differences between using De Moivre's theorem and the sum of angles formulas?
5. How do binomial coefficients play a role in expanding trigonometric identities?

**Tip:** The binomial theorem is highly useful in expanding complex expressions, especially when combined with De Moivre’s theorem for trigonometric identities.

Use De Moivre's theorem, together with the binomial theorem, to show that:
1. cos(4θ) = cos⁴(θ) - 6 cos²(θ)sin²(θ) + sin⁴(θ)
2. sin(4θ) = 4 cos³(θ)sin(θ) - 4 cos(θ)sin³(θ)

Learn how to use De Moivre's theorem along with the binomial theorem to derive key trigonometric identities for cos(4θ) and sin(4θ). This step-by-step solution covers advanced trigonometric and complex number concepts, making it suitable for Grades 11-12.

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