Simplify each expression.
± √[(1 + cos (x/4))/2]

Verified step by step guidance

1. Recognize that the expression inside the square root, \(\frac{1 + \cos\left(\frac{x}{4}\right)}{2}\), matches the form of the half-angle identity for cosine: \(\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos(\theta)}{2}\).
2. Identify \(\theta\) such that \(\theta = \frac{x}{2}\), so that \(\frac{\theta}{2} = \frac{x}{4}\). This means the expression inside the root is \(\cos^2\left(\frac{x}{4}\right)\).
3. Rewrite the square root expression as \(\pm \sqrt{\cos^2\left(\frac{x}{4}\right)}\).
4. Since the square root of a square is the absolute value, simplify to \(\pm \left| \cos\left(\frac{x}{4}\right) \right|\).
5. Finally, consider the \(\pm\) sign and the absolute value to write the simplified form as \(\pm \cos\left(\frac{x}{4}\right)\), noting that the sign depends on the context or domain of \(x\).

Key Concepts

Half-Angle Identity for Cosine

The half-angle identity expresses the cosine of half an angle in terms of the cosine of the original angle: cos(θ/2) = ±√[(1 + cos θ)/2]. This formula helps simplify expressions involving cosines of fractional angles by rewriting them in a square root form.

Simplifying Square Root Expressions

Simplifying square root expressions involves recognizing perfect squares and applying algebraic manipulation to reduce the expression to its simplest form. Understanding when to apply the ± sign is crucial, as it depends on the angle's quadrant or domain.

Trigonometric Function Domains and Signs

The ± sign in trigonometric identities depends on the angle's quadrant, which determines the sign of the trigonometric function. Knowing the domain of x/4 helps decide whether the positive or negative root applies, ensuring the correct simplification.