https://file.aiquickdraw.com/mathbot/file/17287546962521kxc1vle.jpg

The image contains three trigonometric identities that need to be verified:

1. \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)
2. \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
3. \( \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \)

There is also an expression derived for \( \cos(3\theta) \):
\[
4\cos^3(\theta) = \cos(3\theta) + 3\cos(\theta)
\]
which implies:
\[
\cos^3(\theta) = \frac{\cos(3\theta)}{4} + \frac{3\cos(\theta)}{4}
\]

Let’s work through these step-by-step.

### (a) Prove \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)

Using the double angle identity for cosine:
\[
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\]
We know that \( \sin^2(\theta) = 1 - \cos^2(\theta) \), so:
\[
\cos(2\theta) = \cos^2(\theta) - (1 - \cos^2(\theta)) = 2\cos^2(\theta) - 1
\]
This verifies the identity.

### (b) Prove \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

The double angle identity for sine is:
\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]
This is a direct application of the identity, so it is already verified.

### (c) Prove \( \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \)

We use the triple angle identity for cosine:
\[
\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)
\]
This is already the given form, so it holds true.

### Derived Expression:
From the expression \( \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \), we rearrange it to get:
\[
4\cos^3(\theta) = \cos(3\theta) + 3\cos(\theta)
\]
Dividing by 4:
\[
\cos^3(\theta) = \frac{\cos(3\theta)}{4} + \frac{3\cos(\theta)}{4}
\]
This step also holds true as shown on the board.

Would you like further clarification on any of these steps, or more details?

Here are five related questions:
1. What is the general form of the triple angle identity for sine?
2. How can we derive the identity for \( \cos(2\theta) \) using Euler's formula?
3. What is the geometric interpretation of the double angle identities?
4. How can we use the Pythagorean identity to simplify expressions involving \( \cos(\theta) \) and \( \sin(\theta) \)?
5. How do these trigonometric identities extend to hyperbolic functions?

**Tip:** When working with trigonometric identities, it is often useful to convert everything into terms of sine and cosine before simplifying.

Prove the trigonometric identities for double and triple angles.

Learn how to prove trigonometric identities for double and triple angles, including cos(2θ) = 2cos²(θ) - 1 and cos(3θ) = 4cos³(θ) - 3cos(θ). This guide covers key trigonometric formulas and theorems for advanced high school students.

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