Problem 37 - Algebra and Trigonometry

Question

Use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. $$ \sin ^{2} x \cos ^{2} x $$

Step-by-Step Solution

Answer

The expression \(\sin ^{2} x \cos ^{2} x\) can be simplified to \( \cos ^{2} x - 1 + 2\sin ^2 x - \sin^4 x \), with the help of power-reducing formulas.

Step 1: Identify the power-reducing formulas to use

Here we see that \(\sin ^{2} x \cos ^{2} x\) contains a sine squared and a cosine squared. The power-reducing formulas are \( \sin ^2 x = 1 - \cos ^2 x \) and \( \cos ^2 x = 1 - \sin ^2 x \). These formulas will be used to reduce the power of the trigonometric functions.

Step 2: Apply the power-reducing formula for the sine squared

Replacing \(\sin ^{2} x\) in the given expression using the formula \( \sin ^2 x = 1 - \cos ^2 x \), the expression \(\sin ^{2} x \cos ^{2} x\) simplifies to \((1 - \cos ^2 x) * \cos ^{2} x\).

Step 3: Expand the expression

Multiplying the terms in the previous expression results in \( \cos ^{2} x - \cos ^{4} x\).

Step 4: Apply the power-reducing formula for the cosine squared

Replacing \(\cos ^{2} x\) in \( \cos ^{4} x \) using the formula \( \cos ^2 x = 1 - \sin ^2 x \), the expression simplifies to \( \cos ^{2} x - (1 - \sin ^2 x)^2\).

Step 5: Expand and simplify

After expanding \((1 - \sin ^2 x)^2\) and simplifying, we are left with \( \cos ^{2} x - 1 + 2\sin ^2 x - \sin^4 x \), which is an expression that does not contain powers of trigonometric functions greater than 1.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables. They are fundamental tools in mathematics that help simplify complex problems and expressions. When working with trigonometric expressions, such as \(\sin^2x\cos^2x\), identities come to the rescue by allowing us to rewrite these functions in a different form. One of the key identities used in the given problem is the power-reducing formulas. These are special trigonometric identities that help us reduce the powers of sine and cosine functions.

• \( \sin^2x = 1 - \cos^2x \)
• \( \cos^2x = 1 - \sin^2x \)

These formulas are derived from the Pythagorean identity, \( \sin^2x + \cos^2x = 1 \), which is one of the most important trigonometric identities that relates the square of sine and cosine to unity. By using these identities, we can transform expressions to make them easier to work with.

Trigonometric Simplification

Trigonometric simplification involves rewriting trigonometric expressions to make them simpler or more manageable. In the context of the exercise, the expression \(\sin^2x\cos^2x\) needed simplification to an equivalent form without higher powers of trigonometric functions. We achieved this through the power-reducing formulas.
To simplify means:

• Identifying parts of the expression that can be changed using known identities.
• Replacing these parts with their identities, such as substituting \(\sin^2x\) with \(1 - \cos^2x\).
• Carrying out algebraic operations, like expanding and combining like terms.

Simplification often requires patience and practice, but mastering it can make solving trigonometric problems much more straightforward. It also involves seeing the connections between different parts of an expression and knowing how to manipulate them effectively.

Algebraic Manipulation

Algebraic manipulation is the process of transforming mathematical expressions through operations and transformations. In our exercise, after utilizing trigonometric identities, we faced the task of manipulating the resulting algebraic forms. This is a crucial step for simplifying the expression fully.

• First, we replaced \(\sin^2x\) using \(\sin^2x = 1 - \cos^2x\), leading to the expression \((1 - \cos^2x)\cos^2x\).
• Next, we expanded this as \(\cos^2x - \cos^4x\).
• Then, further substitution replaced \(\cos^2x\) in \(\cos^4x\) using \(\cos^2x = 1 - \sin^2x\), simplifying to \(\cos^2x - (1 - \sin^2x)^2\).
• Finally, we expanded \((1 - \sin^2x)^2\) and simplified to \(\cos^2x - 1 + 2\sin^2x - \sin^4x\).

Each of these steps required careful attention to detail and familiarity with algebraic operations such as expansion, factorization, and simplification. By mastering algebraic manipulation, you gain the ability to handle complex mathematical expressions and equations with ease.