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.