Verify that each equation is an identity.
(1/2)cot (x/2)...

Question

Verify that each equation is an identity.

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity, we're asked whether something is an identity. What that's really asking is, does the equation hold true for any X value? And the equation we're given in this case, we have the factor one half co C can of X divided by two minus one half C can of X divided by two multiplied by the factor cosine of X divided by two plus sine of X divided by two is equal to tangent of X. We have two answer choices. Option A, option B no. Yeah. Well, we're trying to figure out if this is an identity. What we wanna do is work with one of the sides rearrange. It simplify it and try to get it so that we can write it in terms of the right hand side, right? To show that these two sides are equal. In this case, we're gonna start with the left hand side because it's messier, right. So it's gonna be a little bit easier to simplify that one rather than trying to take tangent and break it up into kind of a messy looking version of the left hand. OK. So let's start with the left hand side. And again, that's the factor one half Ock FX divided by two minus one half C can of X divided by two multiplied by the factor cosine of X divided by two plus sine of X divided by two. And the first thing we're gonna do is expand this, OK. So using foil or any other method that you use to expand binomials multiplied together, go ahead and use that. And when we do that, we get the following one half co-signed of X divided by two, multiplied by KC can of X divided by two plus one half nine X divided by two multiplied by kant of X divided by two minus one half cosine of X divided by two, you can X divided by two minus one half nine of X divided by two multiplied by C FX divided by two. OK. So this is a very big long expression, but it's just four terms added together. And some of these are actually gonna simplify right away easily. OK? Starting with science multiplied by Kakc can is one divided by sine. So sine multiplied by Koi Can of the same angle. That's just going to be what? OK. And then we're multiplying by half. So we get a ha and we have the same thing for some of these other terms. Um while we work this all out, OK, coast sign multiplied by CO C can by CC can is one divided by sine, cosine multiplied by one divided by sine that gives us cotangents. So we have one half, tangent of X divided by two less one half minus one half. OK? Because we have cosine multiplied by C camp C can is one divided by cosine. We just get that coefficient out in front there. And then this final term gives us minus one half and we get tangent of X divided by two because we have s multiplied by C can C can is one divided by cosine. So we get sign divided by cosine which we know is equal to tangent. OK. Our plus one half and our minus one half those are gonna add to zero. So all we're left with is one half. So tangent of X divided by two minus one half tangent of X divided by two. Now, on the right hand side of our equation, the angle we have is just X, maybe we have tangent of X. Here, we have an angle of X divided by two. So we wanna use our half angle formulas or double angle formulas one of those to try to get this in terms of the angle just X. So we can see if it's equal to the right hand side. Now we have one tangent, we have one cotangent. Let's try to write this all in terms of tangent. And we'll see why in just a minute. And when we write this, we can write this as one divided by two, multiplied by tangent of X divided by two minus one half tangent of X divided by two. If we write it in this way, we can then find a common denominator. We want that denominator to be two tangent of X divided by two. And we get one minus the second term, we have to multiply by tangent of X divided by two to get that common denominator. So we end up with tangent squared of X divided by two. And in the denominator, we have two tangent of X divided by two. And the reason we've written it like this is because we have a double angle formula and recall that that is cotangent of two, theta is equal to one minus tangent squared, the divided by two tangent theta. So this is the exact same form of the equation we have here. The only difference is that the theta we have is X divided by two. What that means is that two theta is just going to be X. So we have written an equation actually for code tangent of X code tangent of X is going to be equal to one minus tangent squared of X divided by two, divided by two tangent of X divided by two. And that is exactly what we have. OK. So our entire expression is going to be equal to cotangent of X. We managed to simplify this all the way down, which is awesome. Unfortunately, it does not equal the right hand side. And on the left, we have cotangent of X. On the right, we have tangent of X. We know that those two functions are not equal to each other. For every X value. This does not equal to the right hand side. And so this is not an identity. If we compare this to our answer choices, we can see that option B is the correct answer. No, thanks everyone for watching. I hope this video helped you in the next one.

Verify that each equation is an identity.
(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

Key Concepts

Trigonometric Identities

Half-Angle Formulas

Cotangent and Tangent Relationship

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