For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
sec x/csc x

II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x

검증된 단계별 안내

Recall the definitions of the secant and cosecant functions in terms of sine and cosine: \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\).

Rewrite the given expression \(\frac{\sec x}{\csc x}\) by substituting these definitions: \(\frac{\frac{1}{\cos x}}{\frac{1}{\sin x}}\).

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \(\frac{1}{\cos x} \times \frac{\sin x}{1} = \frac{\sin x}{\cos x}\).

Recognize that \(\frac{\sin x}{\cos x}\) is the definition of the tangent function, so the expression simplifies to \(\tan x\).

Therefore, the expression \(\frac{\sec x}{\csc x}\) is equivalent to \(\tan x\), which completes the identity.

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주요 개념

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

Reciprocal Trigonometric Functions

Reciprocal functions relate basic trig functions to their inverses, such as sec x = 1/cos x and csc x = 1/sin x. Understanding these allows rewriting expressions like sec x/csc x in terms of sine and cosine for simplification.

Simplifying Trigonometric Expressions

Simplification involves rewriting expressions using fundamental identities and algebraic manipulation. For example, converting sec x/csc x into (1/cos x) ÷ (1/sin x) helps reduce the expression to a simpler form like sin x/cos x.

Trigonometric Identities

Identities such as tan x = sin x/cos x are essential for recognizing equivalent expressions. After simplification, sec x/csc x can be identified as tan x, completing the identity by matching expressions from both columns.

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.sec x/csc xIIA. sin ^2 x/cos ^2 xB.1/(sec ^2 x)C. sin (-x)D. csc ^2 x-cot ^2 x + sin ^2 xE. tan x