Simplify each expression.
±√[(1 - cos 5A)/(1 + cos 5A)]

검증된 단계별 안내

Recognize that the expression involves a square root of a fraction with trigonometric functions: \(\pm \sqrt{\frac{1 - \cos 5A}{1 + \cos 5A}}\).

Recall the trigonometric identity for tangent in terms of cosine: \(\tan^2 \theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}\). Notice the similarity between this identity and the given expression.

Set \(\theta = \frac{5A}{2}\) so that \(2\theta = 5A\). Substitute into the identity to rewrite the expression inside the square root as \(\tan^2 \left( \frac{5A}{2} \right)\).

Since the square root of \(\tan^2 \left( \frac{5A}{2} \right)\) is \(|\tan \left( \frac{5A}{2} \right)|\), and the original expression includes \(\pm\), the simplified form is \(\pm \tan \left( \frac{5A}{2} \right)\).

Therefore, the original expression simplifies to \(\pm \tan \left( \frac{5A}{2} \right)\), using the tangent half-angle identity.

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Key identities like the Pythagorean identity and angle sum formulas help simplify expressions by rewriting them in more manageable forms.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the cosine or sine of the full angle. For example, sin²(θ/2) = (1 - cos θ)/2, which is useful for simplifying expressions involving ratios like (1 - cos x)/(1 + cos x).

Simplification of Radical Expressions

Simplifying radical expressions involves rewriting square roots in simpler or more recognizable forms, often by factoring or using trigonometric identities. Recognizing patterns under the root can help convert complex radicals into trigonometric functions or simpler expressions.

Simplify each expression.±√[(1 - cos 5A)/(1 + cos 5A)]