In Exercises 1–60, verify each identity. tan x csc x cos x = 1

검증된 단계별 안내

Start by recalling the definitions of the trigonometric functions involved: \( \tan x = \frac{\sin x}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), and \( \cos x = \cos x \).

Substitute these definitions into the left-hand side of the identity: \( \tan x \cdot \csc x \cdot \cos x = \left( \frac{\sin x}{\cos x} \right) \cdot \left( \frac{1}{\sin x} \right) \cdot \cos x \).

Simplify the expression by canceling out \( \sin x \) in the numerator and denominator: \( \frac{\sin x}{\cos x} \cdot \frac{1}{\sin x} = \frac{1}{\cos x} \).

Now, multiply the remaining terms: \( \frac{1}{\cos x} \cdot \cos x \).

Observe that \( \frac{1}{\cos x} \cdot \cos x = 1 \), thus verifying the identity.

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables where both sides of the equation are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.

Reciprocal Functions

Reciprocal functions in trigonometry refer to pairs of functions where one function is the reciprocal of another. For example, cosecant (csc) is the reciprocal of sine (sin), and secant (sec) is the reciprocal of cosine (cos). Recognizing these relationships helps in manipulating and simplifying trigonometric expressions, which is essential for verifying identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic techniques to rewrite expressions in a more manageable form. This process often includes factoring, combining like terms, and substituting equivalent functions. Mastery of simplification techniques is vital for verifying identities, as it allows one to transform one side of the equation to match the other.