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.
-tan x cos 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

Identify the given expression in Column I: \(-\tan x \cos x\).

Recall the definition of tangent in terms of sine and cosine: \(\tan x = \frac{\sin x}{\cos x}\).

Rewrite the expression by substituting \(\tan x\) with \(\frac{\sin x}{\cos x}\): \(-\tan x \cos x = -\left(\frac{\sin x}{\cos x}\right) \cos x\).

Simplify the expression by canceling \(\cos x\) in numerator and denominator: \(-\left(\frac{\sin x}{\cos x}\right) \cos x = -\sin x\).

Conclude that the expression \(-\tan x \cos x\) is equivalent to \(-\sin x\), which can be matched to the corresponding expression in Column II.

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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. They allow rewriting expressions in different but equivalent forms, which is essential for simplifying or matching expressions in problems.

Relationship Between Tangent, Sine, and Cosine

The tangent function can be expressed as the ratio of sine to cosine: tan x = sin x / cos x. This relationship helps in rewriting expressions involving tangent in terms of sine and cosine, facilitating simplification or comparison.

Algebraic Manipulation of Trigonometric Expressions

Algebraic manipulation involves factoring, expanding, or rearranging trigonometric expressions to reveal equivalent forms. This skill is crucial when matching expressions from different columns or proving identities.

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.-tan x cos 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