Problem 52 - Precalculus with Limits

Question

Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \sin^4 x \cos^4 x \)

Step-by-Step Solution

Answer

The expression \(\sin^4(x) cos^4(x)\) can be rewritten in terms of the first power of cos as: \(\frac{3}{64} - \frac{1}{16}\cos(2x) + \frac{1}{32}\cos(4x) + \frac{1}{64}\cos^2(4x)\).

Step 1: Convert to Power-Reducing Identities

First, rewrite the expression using the power-reducing identities: \(\sin^4(x) cos^4(x) = \left(\frac{1 - \cos(2x)}{2}\right)^2 \left(\frac{1 + \cos(2x)}{2}\right)^2\).

Step 2: Simplify the Expression

Expand these terms: \(\left(\frac{1 - \cos(2x)}{2}\right)^2 \left(\frac{1 + \cos(2x)}{2}\right)^2 = \frac{1}{16}(1 - 2\cos(2x) + \cos^2(2x))(1 + 2\cos(2x) + \cos^2(2x))\).

Step 3: Use Power-Reducing Identity for \(\cos^2(2x)\)

Now, use the power-reducing formula for \(\cos^2(2x)\) - \(\frac{1 + \cos(4x)}{2}\). So, replace every \(\cos^2(2x)\) in your equation by \(\frac{1 + \cos(4x)}{2}\), yielding \(\frac{1}{16}\left(1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}\right)\left(1 + 2\cos(2x) + \frac{1 + \cos(4x)}{2}\right)\).

Step 4: Simplify the Equation Further

At this point, further simplifications can be made. Do the multiplications and collect like terms. By doing so, the final simplified form is obtained.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions to one another. These identities are essential for simplifying trigonometric expressions and solving trigonometric equations. There are several categories of these identities, including reciprocal identities, quotient identities, Pythagorean identities, even-odd identities, sum and difference identities, double angle identities, and power-reducing formulas.

The power-reducing formulas, in particular, are used in our example to convert higher powers of sine and cosine functions into expressions involving the first power of the cosine function. The formula for reducing the power of a sine squared term is \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \) and for a cosine squared term is \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \). These identities are derived from the double angle formulas and are invaluable tools for simplifying complex trigonometric expressions, as seen in the given exercise.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves a series of algebraic and trigonometric manipulations to present an expression in a more compact, understandable, or useful form. This is particularly helpful when dealing with integrals, derivatives, or solving trigonometric equations.

In the step-by-step solution provided, the process starts with applying the power-reducing formulas, followed by expanding and simplifying the resulting expressions. Careful expansion and the use of algebraic principles such as distributing and combining like terms are needed to reach a simpler form. Simplification often involves recognizing patterns and applying relevant identities efficiently. To optimize the simplification process, one must be familiar with the basic properties of trigonometric functions and have a strategic approach, such as grouping terms wisely to facilitate cancellation or simplification.

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions. It is defined as the ratio of the adjacent side to the hypotenuse of a right-angled triangle. The cosine function is also defined on the unit circle, where it represents the x-coordinate of a point on the circle corresponding to a given angle.

The properties of the cosine function are crucial for understanding its behavior and its use in various trigonometric identities. For instance, the cosine function is an even function, meaning \( \cos(-x) = \cos(x) \), and it has a period of \( 2\pi \), repeating its values every \( 2\pi \) radians. This function is instrumental in creating the power-reducing formulas used in the textbook exercise example, allowing the conversion of \( \sin^4 x \cos^4 x \) into terms involving the first power of the cosine.