Verify that each equation is an identity.
(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

검증된 단계별 안내

Start by recalling the definitions of cotangent and tangent in terms of sine and cosine: \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Rewrite the left-hand side (LHS) of the equation \(\frac{1}{2} \cot \left( \frac{x}{2} \right) - \frac{1}{2} \tan \left( \frac{x}{2} \right)\) using these definitions: express both cotangent and tangent as fractions involving sine and cosine.

Combine the two terms on the LHS over a common denominator to simplify the expression. This will involve algebraic manipulation of fractions.

Use the double-angle identities for sine and cosine, specifically \(\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}\) and \(\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}\), to rewrite the expression in terms of \(\sin x\) and \(\cos x\).

After simplification, compare the resulting expression to the right-hand side (RHS) \(\cot x = \frac{\cos x}{\sin x}\). If both sides match, the identity is verified.

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주요 개념

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means manipulating one side of the equation using known identities until it matches the other side, confirming their equivalence.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For example, cot(x/2) and tan(x/2) can be related to functions of x, which helps simplify or transform expressions involving half angles.

Cotangent and Tangent Relationship

Cotangent and tangent are reciprocal functions: cot(θ) = 1/tan(θ). Understanding their relationship allows for rewriting expressions involving cot and tan, facilitating simplification and verification of identities.

Verify that each equation is an identity.(1/2)cot (x/2) - (1/2) tan (x/2) = cot x