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 1.6 sin⁴ x

Verified step by step guidance

1. Recognize that the expression involves a power of sine, specifically \( \sin^4 x \).
2. Recall the power-reducing formula for \( \sin^2 x \), which is \( \sin^2 x = \frac{1 - \cos(2x)}{2} \).
3. Express \( \sin^4 x \) as \( (\sin^2 x)^2 \).
4. Substitute the power-reducing formula for \( \sin^2 x \) into the expression: \( (\sin^2 x)^2 = \left( \frac{1 - \cos(2x)}{2} \right)^2 \).
5. Simplify the expression by expanding the square: \( \left( \frac{1 - \cos(2x)}{2} \right)^2 = \frac{(1 - \cos(2x))^2}{4} \).

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 or cosine. For example, the formula for sine squared is sin²(x) = (1 - cos(2x))/2. These formulas are essential for simplifying expressions that contain higher powers of trigonometric functions, allowing for easier manipulation and evaluation.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. They include fundamental identities such as the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for transforming and simplifying trigonometric expressions, especially when applying power-reducing formulas.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions involves rewriting complex expressions into simpler forms, often using identities and formulas. This process is important in trigonometry as it allows for easier calculations and clearer understanding of relationships between angles and their corresponding trigonometric values. Mastery of simplification techniques is vital for solving problems effectively.