Deriving trigonometric identities
b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

검증된 단계별 안내

Step 1: Start with the given identity for \( \cos 2t \), which is \( \cos 2t = 2 \cos^2 t - 1 \).

Step 2: Differentiate both sides of the equation with respect to \( t \). For the left side, use the derivative of \( \cos 2t \), which is \(-2 \sin 2t\) using the chain rule.

Step 3: For the right side, differentiate \( 2 \cos^2 t - 1 \). Use the chain rule to differentiate \( 2 \cos^2 t \), which gives \( 2 \times 2 \cos t \times (-\sin t) = -4 \cos t \sin t \). The derivative of \(-1\) is 0.

Step 4: Set the derivatives from both sides equal to each other: \(-2 \sin 2t = -4 \cos t \sin t\).

Step 5: Simplify the equation \(-2 \sin 2t = -4 \cos t \sin t\) to verify the identity for \( \sin 2t \). Recognize that \( \sin 2t = 2 \sin t \cos t \), confirming the identity.

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities. Understanding these identities is crucial for simplifying expressions and solving equations in trigonometry.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to its variable. In the context of trigonometric functions, differentiation allows us to find the slopes of curves and can be used to verify identities by showing that two expressions yield the same derivative.

Double Angle Formulas

Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the formula for cosine of double angle is cos(2t) = 2cos²(t) - 1. These formulas are essential for simplifying expressions and proving identities, particularly when differentiating or integrating trigonometric functions.

Deriving trigonometric identitiesb. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.