Problem 19 - Precalculus: Functions and Graphs

Question

Verify the identity. \(\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1\)

Step-by-Step Solution

Answer

Identity \( \cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1 \) is verified.

Step 1: Understand the Identity

We need to verify that the expression \( \cos 4 \theta \) is equal to \( 8 \cos^4 \theta - 8 \cos^2 \theta + 1 \). This exercise requires us to use trigonometric identities to transform one side into the other.

Step 2: Use Double Angle Identity for Cosine

Recall the double angle identity: \( \cos 2\alpha = 2 \cos^2 \alpha - 1 \). We'll use this identity to rewrite \( \cos 4 \theta \) in terms of \( \cos^2 2 \theta \).

Step 3: Rewrite \( \cos 4\theta \) Using Double Angle

The identity \( \cos 4\theta = \cos(2 imes 2\theta) \) can be written as \( \cos 4\theta = 2 \cos^2 2\theta - 1 \) by applying the double angle formula.

Step 4: Apply Double Angle to \( \cos^2 2\theta \)

Apply the double angle identity again to \( \cos 2\theta = 2 \cos^2 \theta - 1 \). Thus, \( \cos^2 2\theta = (2 \cos^2 \theta - 1)^2 \).

Step 5: Expand \( (2 \cos^2 \theta - 1)^2 \)

Expand \( (2 \cos^2 \theta - 1)^2 = 4 \cos^4 \theta - 4 \cos^2 \theta + 1 \).

Step 6: Substitute and Simplify

Substitute back into Step 3: \( \cos 4\theta = 2 (4\cos^4\theta - 4\cos^2\theta + 1) - 1 \). Simplify to get \( \cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 2 - 1 \).

Step 7: Final Simplification

Combine the terms to get \( \cos 4\theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1 \), which matches the right side of the original identity.

Key Concepts

Double Angle Formulas

Double angle formulas in trigonometry help us express trigonometric functions of angles like \(2\alpha\) in terms of \(\alpha\). They are incredibly useful for simplifying expressions and solving equations more easily, especially when you work with angles that are double another. For instance, with the cosine function, the double angle formula is given as:

• \(\cos 2\alpha = 2 \cos^2 \alpha - 1\)

This means instead of dealing directly with \(\cos 2\alpha\), we can rewrite it in terms of \(\cos^2 \alpha\).
By utilizing these formulas, we can rewrite more complicated functions like \(\cos 4\theta \) by recognizing it as \( \cos (2 \times 2\theta) \). Applying the double angle formula twice, as shown in the solution, allows us to verify identities by breaking down larger angles into smaller, more manageable ones.

Cosine Function

The cosine function is one of the primary trigonometric functions and is crucial in both pure and applied mathematics. It is fundamentally a ratio of the adjacent side to the hypotenuse in a right triangle. However, in more advanced math like calculus and beyond, we see it appear in functional forms such as:

• \(\cos \theta\), which varies between -1 and 1 as \(\theta\) changes. It's a periodic function with a period of \(2\pi\).

In problems like verifying the given identity, understanding how the cosine behaves, especially under transformation by identities or formulas, is essential.
Whenever you see expressions like \(8\cos^4\theta\), know that they refer to raising the cosine of a particular angle to a power, altering the function's amplitude and shape. Recognizing how these powers of cosine contribute to the overall identity verification process is a skill developed through practice and familiarity with trigonometric identities.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to help solve equations or prove identities. It is a fundamental tool in mathematics. Within the context of trigonometric identities, like the one we are discussing, algebraic manipulation brings several steps together.

• Start by using known identities to transform expressions.
• Expand expressions, like \((2 \cos^2 \theta - 1)^2\), into more straightforward forms.
• Simplify by combining like terms or factoring common components.

In solving our identity, the steps:

1. Substituting identities.
2. Expanding expressions.
3. Carefully simplifying each term.

are critical to reducing the problem into something that visibly matches the right side of the identity. By understanding these principles of algebraic manipulation, you get more efficient at tackling more complex identities and equations across various areas of mathematics.