Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]

Verified step by step guidance

1. Recognize that the expression involves the square root of a fraction with cosine terms: \(\sqrt{\frac{1 + \cos 165^\circ}{1 - \cos 165^\circ}}\).
2. Recall the trigonometric identity for tangent in terms of cosine: \(\tan^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{1 + \cos \theta}\), which can be rearranged to relate to the given expression.
3. Rewrite the expression inside the square root by comparing it to the identity: \(\frac{1 + \cos 165^\circ}{1 - \cos 165^\circ} = \frac{1}{\frac{1 - \cos 165^\circ}{1 + \cos 165^\circ}}\) and recognize this as the reciprocal of \(\tan^2 \frac{165^\circ}{2}\).
4. Simplify the square root of the reciprocal of \(\tan^2 \frac{165^\circ}{2}\) to get \(\frac{1}{\tan \frac{165^\circ}{2}}\), which is equal to \(\cot \frac{165^\circ}{2}\).
5. Evaluate or express the final simplified form as \(\cot 82.5^\circ\), which is the simplified trigonometric expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle values. Key identities like the Pythagorean identity and angle sum/difference formulas help simplify expressions by rewriting complex terms into simpler or more familiar forms.

Cosine of Special Angles

Knowing the exact values of cosine for special angles, such as 165°, is essential. Since 165° can be expressed as 180° - 15°, the cosine difference identity allows calculation of cos 165° using cos(180° - θ) = -cos θ, facilitating simplification.

Simplification of Radical Expressions

Simplifying expressions involving square roots often requires rationalizing or rewriting the radicand using algebraic or trigonometric identities. Recognizing patterns like (1 + cos θ)/(1 - cos θ) can be transformed using half-angle or other identities to simplify the radical.