Verify that each equation is an identity.
2 cos² (x/2) ...

Question

Verify that each equation is an identity.

2 cos² (x/2) tan x = tan x+ sin x

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity. And when we, we're asked this question, what that means is, does this equation hold true for any X value? The equation? We have two sine squared of X divided by two multiplied by cotangent of X is equal to cotangent of X plus cosine of X. We have two answer choices. Option A yes and option B no. So let's get started and we're gonna start with the left hand side of this equation. And we're gonna see if we can rewrite this to look like the right hand side. So we have two sine squared of X divided by two multiplied by cotangent of X. We have this kind of half angle, double angle thing happening here. We have one trig function that's using X divided by 21 trig function that's using X. So let's think about how we can rewrite them to be the same. We want the trig functions to be in terms of X not X divided by two because that's what's on the right hand side of our function. So recall the cosine of two theta in our double angle formula is equal to one minus two times squared of the right. OK. Well, we have this two sign squared of something bit. OK. So we want the theta to be our X divided by two, which means that two theta would be X. OK. So if we use this formula, we can write that cosine of X is going to be equal to one minus two S squared of X divided by two. and again, two sine squared X divided by two. That's what we have in that first term. So let's rearrange for that. Specifically, we have two sign square of X divided by two is going to be equal to one minus cosine of X. OK. So we've used our double angle identity, OK. And we thought about doing that because we have X divided by two. In some cases, we have X. In other cases, we wanna get these into common terms. Let's substitute what we found. So now we have one minus cosine of X multiplied by co tangent of X. We can go ahead and expand. And when we expand, we have cotangent of X minus cosine of X, a tangent of X. And we can see that this is not going to be equal to the right hand side. OK. The first term is the first term we have cotangent of X. But then we have minus cosine of X code tangent of X, which is not the same as plus cosine of X OK. There's nothing we can do no identities, we can use those two things are not equal. And so because the left hand side does not equal to the right hand side, this is not an identity. So we have option B now. That's it for this one. Thanks everyone for watching. See you in the next video.

Verify that each equation is an identity.
2 cos² (x/2) tan x = tan x+ sin x

Key Concepts

Trigonometric Identities

Double Angle Formulas

Tangent Function and Its Relationship

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