Deriving trigonometric identities
a. Differentiate both sides of the identity cos 2t = cos² t−sin² t to prove that sin 2 t= 2 sin t cos t.

Verified step by step guidance

1. Step 1: Differentiate the left side of the identity with respect to t. The left side is \( \cos(2t) \). Using the chain rule, the derivative is \(-2\sin(2t)\).
2. Step 2: Differentiate the right side of the identity with respect to t. The right side is \( \cos^2(t) - \sin^2(t) \). Use the chain rule and the power rule to differentiate each term separately.
3. Step 3: For \( \cos^2(t) \), use the chain rule: the derivative is \( 2\cos(t)(-\sin(t)) = -2\cos(t)\sin(t) \).
4. Step 4: For \( \sin^2(t) \), use the chain rule: the derivative is \( 2\sin(t)\cos(t) \).
5. Step 5: Combine the derivatives from steps 3 and 4: \(-2\cos(t)\sin(t) - 2\sin(t)\cos(t) = -2\sin(t)\cos(t) - 2\sin(t)\cos(t) = -4\sin(t)\cos(t)\). Set this equal to the derivative from step 1, \(-2\sin(2t)\), and simplify to show \( \sin(2t) = 2\sin(t)\cos(t) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They serve as fundamental tools in calculus and can simplify complex expressions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which are essential for manipulating and proving relationships between sine and cosine functions.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of trigonometric functions, differentiation applies specific rules, such as the derivatives of sine and cosine, to derive new relationships. This process is crucial for proving identities by showing that two expressions have 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 sin(2t) = 2sin(t)cos(t) is derived from the sine and cosine functions. Understanding these formulas is essential for simplifying expressions and proving identities, as they provide a direct relationship between angles and their trigonometric values.