Verify that each equation is an identity.
tan α/sec α = sin α

검증된 단계별 안내

Start by expressing \( \tan \alpha \) and \( \sec \alpha \) in terms of \( \sin \alpha \) and \( \cos \alpha \). Recall that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) and \( \sec \alpha = \frac{1}{\cos \alpha} \).

Substitute these expressions into the left side of the equation: \( \frac{\tan \alpha}{\sec \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos \alpha}} \).

Simplify the expression by multiplying the numerator by the reciprocal of the denominator: \( \frac{\sin \alpha}{\cos \alpha} \times \cos \alpha \).

Notice that the \( \cos \alpha \) terms cancel each other out, leaving \( \sin \alpha \).

Conclude that the left side simplifies to \( \sin \alpha \), which matches the right side of the equation, thus verifying the identity.

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 quotient identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Tangent and Secant Functions

The tangent function, tan(α), is defined as the ratio of the sine and cosine functions: tan(α) = sin(α)/cos(α). The secant function, sec(α), is the reciprocal of the cosine function: sec(α) = 1/cos(α). Recognizing these relationships is essential for manipulating and verifying trigonometric equations.

Sine Function

The sine function, sin(α), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is a fundamental trigonometric function that plays a key role in various identities and equations. Understanding how sine interacts with other trigonometric functions is vital for verifying identities.

Verify that each equation is an identity.tan α/sec α = sin α