Verify that each equation is an identity.
sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

검증된 단계별 안내

Step 1: Start by simplifying the left-hand side (LHS) of the equation: \( \frac{\sin \theta}{1 - \cos \theta} - \frac{\sin \theta \cos \theta}{1 + \cos \theta} \).

Step 2: Find a common denominator for the fractions on the LHS: \((1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta\).

Step 3: Rewrite each fraction with the common denominator: \( \frac{\sin \theta (1 + \cos \theta) - \sin \theta \cos \theta (1 - \cos \theta)}{\sin^2 \theta} \).

Step 4: Simplify the numerator: \( \sin \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + \sin \theta \cos^2 \theta = \sin \theta + \sin \theta \cos^2 \theta \).

Step 5: Factor out \( \sin \theta \) from the numerator: \( \frac{\sin \theta (1 + \cos^2 \theta)}{\sin^2 \theta} = \csc \theta (1 + \cos^2 \theta) \), which matches the right-hand side (RHS).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations as identities.

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their respective cosecant, secant, and cotangent functions. For example, csc θ is the reciprocal of sin θ, defined as 1/sin θ. Recognizing these relationships helps in transforming and simplifying expressions, particularly when verifying identities.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, combining like terms, and applying common denominators. Mastery of these techniques is essential for verifying trigonometric identities, as it allows one to transform one side of the equation to match the other.

Verify that each equation is an identity.sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)