Deriving trigonometric identities
b. Verify that you o...

Question

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.

Accepted Answer

Welcome back everyone. Which of the following identities is verified by differentiating both sides of sin squared of x equals 1 minus cosine of 2 x divided by 2? We're given 4 answer choices A sin square x plus cosine square x equals 1, B cosine square x equals 1 plus cosine of 2 x divided by 2. C cosine of 2 x equals cosine square x minus 2 X, and D sin of 2 x equals 2 of x cosine of X. So we want to differentiate both sides of the given identity. Let's begin with sin squared of X. So we want to find its derivative. This is the left hand side of the equation, and this is a composite function that consists of an exponent. So we have to recall the power rule. If we differentiate X to the power of N, we get N multiplied by X to the power of N minus 1. Applying this rule, we have to bring down the exponent. We get 2. Sine X because now the exponent becomes one, and then because this is a composite composite function, we need to multiply by the derivative of the inner function, which is the derivative of sine. Based on the table of basic derivatives, the derivative of sine x is cosine x. So we got two sin Xs. Multiplied by Cosine of X. Well done, we have the first derivative. Now, for the second derivative, we want to differentiate 1 minus cosine of 2 Xs divided by 2. And before we find the derivative, well, we can factor out the constant. It's 1/2, so we are finding the derivative of 1 minus cosine of 2 X. We call that the derivative of a constant is 0, so the derivative of 1 becomes 0, and we get 1/2 multiplied by the derivative of negative cosine 2 X. So we can factor out the negative sign and evaluate the derivative of cosine of 2 X. Once again, this is a composite function, so we get -12. The derivative of cosine is negative signs. We get negative sign of 2 X. And based on the chain rule we multiply that by the derivative of the inner function which is to X a game. So combining the results negative, multiplied by negative gives us positive. And then one half multiplied by 2 gives us 1. So we get sign. Of 2 X, which is our final result, right? And this is because the derivative of 2 X is simply 2. And 2 multiplied by 12 gives us 1. So essentially we now see that we have verified the identity such that sin to X equals our previous derivative to sin x. Cosine x which essentially corresponds to the answer choice D. Thank you for watching.

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.

Key Concepts

Trigonometric Identities

Differentiation

Double Angle Formulas

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