Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]

Verified step by step guidance

1. Recognize that the expression \( \pm \sqrt{\frac{1 - \cos 8\theta}{1 + \cos 8\theta}} \) can be simplified using trigonometric identities.
2. Recall the identity for tangent in terms of sine and cosine: \( \tan^2 \theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta} \).
3. Notice that the expression \( \frac{1 - \cos 8\theta}{1 + \cos 8\theta} \) resembles \( \tan^2 \theta \) with \( \theta = 4\theta \).
4. Use the identity to rewrite the expression as \( \tan^2 4\theta \).
5. The expression simplifies to \( \pm \tan 4\theta \), considering the square root and the \( \pm \) sign.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Understanding these identities is crucial for simplifying expressions involving trigonometric functions, such as the one presented in the question.

Cosine Function Properties

The cosine function, denoted as cos(θ), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. Its properties, such as the range of values (from -1 to 1) and periodicity (period of 2π), are essential for manipulating expressions involving cosine, particularly when simplifying expressions like (1 - cos 8θ) and (1 + cos 8θ).

Square Root Simplification

Simplifying square roots involves reducing the expression under the square root to its simplest form. In trigonometry, this often includes factoring expressions or using identities to rewrite them in a more manageable way. For the expression ±√[(1 - cos 8θ)/(1 + cos 8θ)], recognizing how to manipulate the numerator and denominator is key to achieving a simplified result.