Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)

검증된 단계별 안내

Start by recalling the double-angle identity for cosine: \(\cos 2x = \cos(x + x) = \cos^2 x - \sin^2 x\).

Express \(\cot x\) in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).

Rewrite the right-hand side of the equation \(\frac{\cot^2 x - 1}{\cot^2 x + 1}\) by substituting \(\cot^2 x\) with \(\frac{\cos^2 x}{\sin^2 x}\), giving \(\frac{\frac{\cos^2 x}{\sin^2 x} - 1}{\frac{\cos^2 x}{\sin^2 x} + 1}\).

Simplify the complex fraction by multiplying numerator and denominator by \(\sin^2 x\) to eliminate the denominators inside the fraction, resulting in \(\frac{\cos^2 x - \sin^2 x}{\cos^2 x + \sin^2 x}\).

Use the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\) to simplify the denominator, so the expression becomes \(\cos^2 x - \sin^2 x\), which matches the double-angle identity for \(\cos 2x\), verifying the identity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Double-Angle Identity for Cosine

The double-angle identity expresses cos 2x in terms of functions of x, such as cos 2x = cos² x - sin² x or cos 2x = 2 cos² x - 1. This identity helps rewrite trigonometric expressions involving 2x into simpler forms involving x, facilitating verification of equations.

Cotangent and Its Relationship to Sine and Cosine

Cotangent is defined as cot x = cos x / sin x. Understanding this relationship allows conversion of expressions involving cot² x into sine and cosine terms, which is essential for manipulating and simplifying the given equation.

Trigonometric Identities and Algebraic Manipulation

Verifying identities requires applying known trigonometric identities and algebraic techniques such as factoring, common denominators, and substitution. This process transforms one side of the equation to match the other, confirming the identity's validity.

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)cos 2x = (cot² x - 1)/(cot² x + 1)