Simplify each expression.
±√[(1 + cos 18x)/2]

Verified step by step guidance

1. Recognize that the expression inside the square root, \(\frac{1 + \cos 18x}{2}\), matches the form of the half-angle identity for cosine: \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\).
2. Identify the angle \(\theta\) such that \(2\theta = 18x\), which gives \(\theta = 9x\).
3. Rewrite the expression inside the square root using the half-angle identity: \(\frac{1 + \cos 18x}{2} = \cos^2 9x\).
4. Take the square root of \(\cos^2 9x\), which results in \(|\cos 9x|\) because the square root of a square is the absolute value of the original expression.
5. Include the \(\pm\) sign given in the original expression, so the simplified form is \(\pm \cos 9x\), noting that the absolute value and \(\pm\) together cover all cases.

Key Concepts

Half-Angle Identity for Cosine

The half-angle identity expresses the cosine of an angle in terms of the cosine of half that angle. Specifically, cos²(θ) = (1 + cos 2θ)/2, which can be rearranged to find expressions involving square roots. This identity is essential for simplifying expressions like √[(1 + cos 18x)/2].

Square Root and ± Sign in Trigonometric Expressions

When taking the square root of a squared trigonometric function, the result includes a ± sign to account for both positive and negative roots. This is important because trigonometric functions can be positive or negative depending on the angle's quadrant, affecting the simplification outcome.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves applying identities and algebraic manipulation to rewrite expressions in simpler or more recognizable forms. Recognizing patterns like half-angle formulas helps reduce complex expressions to basic trigonometric functions, facilitating easier evaluation or further use.