Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Express in terms of the cosine function with exponent \(1 .\) $$ \cos ^{4} 2 x $$

Step-by-Step Solution

Answer

\( \cos^4(2x) = \frac{3}{8} + \frac{1}{2}\cos(4x) + \frac{1}{8}\cos(8x) \).

Step 1: Understand the Expression

We are given the trigonometric function \( \cos^{4}(2x) \). Our task is to express this with an exponent of 1, meaning we want to reframe the expression so that it is a simple cos function where possible.

Step 2: Use Power-Reduction Formula

The power-reduction formulas help in expressing higher powers of trigonometric functions in terms of the first power. The power-reduction formula for cosine is: \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \).

Step 3: Apply the Power-Reduction Formula

First, express \( \cos^4(2x) \) using the power-reduction formula:\[ \cos^4(2x) = (\cos^2(2x))^2. \]

Step 4: Substitute using Power-Reduction Formula

Substitute \( \cos^2(2x) \) using the power-reduction formula:\[ \cos^2(2x) = \frac{1 + \cos(4x)}{2}. \] Thus,\[ \cos^4(2x) = \left(\frac{1 + \cos(4x)}{2}\right)^2. \]

Step 5: Simplify the Expression

Expand and simplify the expression:\[ \cos^4(2x) = \frac{(1 + \cos(4x))^2}{4} = \frac{1 + 2\cos(4x) + \cos^2(4x)}{4}. \] Then, replace \( \cos^2(4x) \) using the power-reduction formula again:\[ \cos^4(2x) = \frac{1 + 2\cos(4x) + \frac{1 + \cos(8x)}{2}}{4}. \]

Step 6: Final Expression

Combine and simplify:\[ \cos^4(2x) = \frac{1 + 2\cos(4x) + \frac{1}{2} + \frac{1}{2}\cos(8x)}{4} \] \[ \cos^4(2x) = \frac{3}{8} + \frac{1}{2}\cos(4x) + \frac{1}{8}\cos(8x). \] This expresses \( \cos^4(2x) \) in terms of \( \cos \) with an exponent of 1.

Key Concepts

Power-Reduction Formula

The power-reduction formula is a trigonometric identity that allows us to rewrite expressions involving higher powers of sine and cosine in terms of lower powers. This is particularly useful when simplifying expressions or solving integrals. The power-reduction formula for cosine states:

• \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \)

This formula reduces the power of the cosine function from two to one. In the exercise, we want to simplify \( \cos^4(2x) \) to have an exponent of 1. By applying the power-reduction formula twice, we decompose \( \cos^4(2x) \) by first expressing it as \( (\cos^2(2x))^2 \) and then further simplifying it using the formula.
The focus is to iteratively apply the formula to reduce the power, which eventually gives us an expression in terms of simple cosine functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are indispensable tools when it comes to simplifying trigonometric expressions. In the given exercise, multiple identities are used to break down a complex expression into simpler components.
Key to solving problems like this involves recognizing which identities can be strategically applied.

• The power-reduction identity helps manage and reduce the power of trigonometric functions.
• The angle-doubling property is inherent in these identities, helping transition from \( \cos^2(\theta) \) to \( 1 + \cos(2\theta) \).

By understanding these identities, it becomes straightforward to substitute and rearrange expressions to achieve a simpler form. This methodical use of identities ensures that any higher powers can be consistently expressed in terms of a single occurrence of the trigonometric function.

Simplification Steps

In mathematics, simplification refers to the process of making an expression as simple as possible. This may involve combining like terms, reducing fractions, or applying mathematical identities, like those involving trigonometry. For the expression \( \cos^4(2x) \), we aim to simplify it down to functions involving only \( \cos(4x) \) and \( \cos(8x) \) with exponents of 1.
The steps taken to simplify involve:

• Initial Expression Transformation: The original power of four is recognized. By using the identity \( (\cos^2(2x))^2 \), it starts breaking down.
• Applying Identical Substitutions: Insert the power-reduction formula into the squared expression.
• Combining and Distributing: Expand the newly formed square and simplify by combining like terms.

Finally, replacing any remaining power-two terms with their identity and ensuring only terms with the first power of cosine remain. This strategic breakdown not only simplifies computations but also aids in clearer understandings, aligning complex trigonometric forms into manageable expressions.