In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.sin² x cos² x

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1. Identify the power-reducing formulas for \( \sin^2 x \) and \( \cos^2 x \): \( \sin^2 x = \frac{1 - \cos(2x)}{2} \) and \( \cos^2 x = \frac{1 + \cos(2x)}{2} \).
2. Substitute the power-reducing formulas into the expression \( \sin^2 x \cos^2 x \).
3. The expression becomes \( \left( \frac{1 - \cos(2x)}{2} \right) \left( \frac{1 + \cos(2x)}{2} \right) \).
4. Simplify the expression by multiplying the two fractions: \( \frac{(1 - \cos(2x))(1 + \cos(2x))}{4} \).
5. Use the difference of squares formula: \( (a - b)(a + b) = a^2 - b^2 \), to simplify further to \( \frac{1 - \cos^2(2x)}{4} \).

Key Concepts

Power-Reducing Formulas

Power-reducing formulas are trigonometric identities that express powers of sine and cosine in terms of the first power of sine and cosine. These formulas are particularly useful for simplifying expressions involving sin²(x) and cos²(x). For example, sin²(x) can be rewritten as (1 - cos(2x))/2, and cos²(x) can be expressed as (1 + cos(2x))/2.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying trigonometric expressions and solving equations. Key identities include the Pythagorean identities, reciprocal identities, and co-function identities, which provide relationships between different trigonometric functions.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting complex expressions into simpler forms, often using identities. This process is essential for solving trigonometric equations or integrating trigonometric functions. By applying power-reducing formulas, one can reduce the degree of the trigonometric functions, making calculations more manageable.