In Exercises 1–60, verify each identity. cot² t /csc t = csc t - sin t

검증된 단계별 안내

Start with the left-hand side (LHS) of the identity: \( \frac{\cot^{2} t}{\csc t} \). Recall the definitions of cotangent and cosecant in terms of sine and cosine: \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \).

Rewrite \( \cot^{2} t \) as \( \left( \frac{\cos t}{\sin t} \right)^{2} = \frac{\cos^{2} t}{\sin^{2} t} \). Substitute this and \( \csc t = \frac{1}{\sin t} \) into the LHS to get \( \frac{\frac{\cos^{2} t}{\sin^{2} t}}{\frac{1}{\sin t}} \).

Simplify the complex fraction by multiplying numerator and denominator: \( \frac{\cos^{2} t}{\sin^{2} t} \times \sin t = \frac{\cos^{2} t \cdot \sin t}{\sin^{2} t} = \frac{\cos^{2} t}{\sin t} \).

Now, focus on the right-hand side (RHS): \( \csc t - \sin t = \frac{1}{\sin t} - \sin t \). To combine these terms, write \( \sin t \) as \( \frac{\sin^{2} t}{\sin t} \) to get a common denominator: \( \frac{1}{\sin t} - \frac{\sin^{2} t}{\sin t} = \frac{1 - \sin^{2} t}{\sin t} \).

Use the Pythagorean identity \( 1 - \sin^{2} t = \cos^{2} t \) to rewrite the numerator, so the RHS becomes \( \frac{\cos^{2} t}{\sin t} \), which matches the simplified LHS, thus verifying the identity.

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주요 개념

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Verifying an identity means showing both sides of the equation are equivalent by using known identities and algebraic manipulation.

Reciprocal and Quotient Identities

Reciprocal identities relate sine, cosine, and tangent to their reciprocal functions cosecant, secant, and cotangent. For example, csc t = 1/sin t and cot t = cos t / sin t. These identities help rewrite expressions to simplify or verify equations.

Algebraic Manipulation of Trigonometric Expressions

Simplifying or verifying identities often requires factoring, combining fractions, and using common denominators. Careful algebraic steps allow transforming one side of the equation into the other, confirming the identity's validity.