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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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