Deriving trigonometric identities
c. Differentiate both ...

Question

Deriving trigonometric identities

c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Accepted Answer

Welcome back everyone to another video. Determine the derivative of the right hand side of the identity tangent of 2 X equals 2 tangent Xs divided by 1 minus tangent squared X. We're given 4 answer choices A says 1 minus tangent squared X squared divided by square x, b 1. A tangent squared of x squared divided by 2 sequences the power of 4th of X C 2 x divided by 1 minus 12 x squared and D says 2 sequences the power of 4th of x divided by 1 minus tent square x2. So now we want to differentiate the right hand side of the given identity. And this means that we are differentiating a 2 tangent of X divided by 1 minus tangent squared of x and for this problem we can recall the quotient rule. Now according to the quotient rule, we have two functions. We have the top one which is F, and that's 2 tangent of X. And we have the bottom one. Let's call it G, right? And now our G is 1 minus tangent squared of x. The quotient rule says that if we want to differentiate F divided by G, this means that we're taking F multiplying by G and subtracting F multiplied by G. All of that is going to be divided by G2, and this means that we need two derivatives. We need F so we can factor out the 2 and the derivative of tangent of X based on. The table of basic derivatives is going to be c2, so we get 2 c 2d of X for our first derivative. For the second derivative, we're taking G, and this means that we're taking the derivative of 1 minus tangent squad of X. The derivative of 1 is 0. Then we can factor out the negative sign, and then we have a composite function. So first of all, we apply. The power rules we bring down the 2. The new exponent of tangent is once we get -2 tangent of X, and based on the chain rule, we now have to multiply by the derivative of the inner function which is tangent of X, and we know that the derivative is C2d of X from the previous step. At this point we have all the derivatives that we need and we can substitute. So now what we're going to do is write our numerator. It begins with F and we can write 2c 2 X. Multiplied by one. Minus tension squared of X, right, because that's our g, so 1 minus. The engine squared off X. And then we are subtracting FG, so our F is 2 tangent of X and our G is -2. Tangent acts. Multiplied by c squared of X. So we have our numerator for the denominator. We simply want to take G2d, and that's 1 minus tangent squad of x, all of that squared. And now we can continue and we can say that this is going to be equal to let's simplify the numerator and we are going to do that by distributing the terms we get to c2d of x minus. 2 squared of X. Tension squared. Of X minus, and now we are actually saying plus because we have -2 multiplied by -2, which gives us +42 squared of X. Tangent squared of x and from here we're going to combine the like terms. Before that, let's write the denominator 1 minus tangent squared of x, all of that squared and now combining the like terms we're going to get 2 so we can immediately factor out the 2 and we're going to get 22 squared of x. And now we can also factor out squad, right? So overall, let's say that we're factoring out 2 squad of X, which gives us 1 + 2 because -2 + 4 gives us 2. We're factoring out the c 2d and this gives us. Tension squared of X and since we're factoring out the 2 well we shouldn't have 2, we will have 1 tangent squared. Of X And that we are dividing that by 1 minus tangent squared of x squad. Now let's recall trigonometric identities, and we're going to say that 1 plus tangent squared X can be rewritten as squared X, right? So this comes from. Trigonometric identities and then we're multiplying 2 squared by another squad, which gives us 2 sequences the power of 4th of X. Divided by 1 minus tent squared of x, all of that squared, which essentially corresponds to the answer choice d. Let's label it as our final answer and thank you for watching.

Deriving trigonometric identities
c. Differentiate both sides of the identity sin 2t = 2 sin t cost to prove that cos 2t = cos²t−sin²t.

Key Concepts

Trigonometric Identities

Differentiation

Double Angle Formulas

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