Deriving trigonometric identities
a. Differentiate both ...

Question

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.

Accepted Answer

Welcome back, everyone. Which of the following identities is verified by differentiating both sides of cosine squared X equals 1 plus cosine of 2 x divided by 2? We were given 4 answer choices. A sin 2 x equals 2 sin x cosine x. B cosine 2 x equals cosine quad x minus 2 X C square x equals 1 minus cosine of 2 x divided by 2 and D square x plus cosine quad x equals 1. So essentially what we want to do is find two derivatives. We want to find the derivative of the left hand side, so we want to differentiate cosine squared X to begin with. Now notice that this is a composite function. First of all, we have cosine squared, right. We will have to apply the power rules. We have to bring down the exponent. We have to write 2, which leaves us with cosines the power of first. So we get 2 cosine of X. We call that x to the power of n is equal to n multiplied by x the power of n minus 1. So we are applying this rule, but now according to the chain rule, we have to multiply that by the derivative of the inner function which is cosine itself. So we're multiplying by the derivative of cosine. According to the table of basic derivatives, the derivative of cosine is negative sign, so we get -2 x x cosine x, and this gives us the left hand side of the equation in the differentiated form. Now for the second derivative, we want to analyze 1 plus cosine of 2 x divided by 2. And we want to find its derivative. So what we can do is first we'll factor out 1/2 and then differentiate 1 plus cosine of 2 Xs. Now we have a sum of two terms, right? So we are going to factor out one half, and we have to recall that the derivative of 1. is equal to 0 because one is a constant. So we simply want to evaluate the derivative of cosine. Off to X. Once again, we will need to apply the chain rule because we have a composite function. First of all, the outer function is the cosine, and we are evaluating the derivative of cosine, which is a negative sign. So we get negative. Sign of two Xs, and then based on the chain rule, we have to multiply that by the derivative of the inner function, which is 2 X. Now this gives us 1/2 multiplied by negative sign of 2 X, so we can put the negative sign in front, and then we get sign. 2 Xs, and the derivative of 2 Xs based on the power rule is just 2. We're multiplying that by 2. Now, let's simplify -12 multiplied by 2 is negative, -1, and then we get negative sign of 2 X. This is our right hand side. So now if we first will write the left hand side, which is negative sign of 2 X, this must be equal to the first derivative that we calculated, and the first derivative was negative to. Sine, cosine x. Notice that we can multiply both sides by -1, which eliminates the negative sign, and we have verified the identity of 2 X equals 2 sin x cosine x, which corresponds to the answer choice A. Thank you for watching.

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.

Key Concepts

Trigonometric Identities

Differentiation

Double Angle Formulas

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