 In this video, we're gonna prove the trigonometric identity sine squared of x over two is equal to tangent x minus sine x over two tangent of x. Well, how are we gonna prove this? Well, I usually like to pick the more complicated side and it's very tempting to think that the right-hand side's more complicated. You have some fractions and differences going on, but I'm actually gonna start with the left-hand side because of the difference of the angles, right? What I mean by that is the left-hand side has an x over two. The right-hand side, all of the angles are x. So the only way we can prove this identity here is we have to convert from x to x over two or the other way around. And so that's where the half-angle identity comes into play here. So remember, when it comes to sine, sine of x over two, this is equal to plus or minus the square root of one minus cosine of x all over two. But this is the half-angle identity for sine, but if you square things, things get a lot easier. You get sine squared of x over two is equal to just one-half one minus cosine of x. And so that's the version that we actually wanna use right here. So let's prove our identity now. Let's start with the left-hand side. Like I said, the left-hand side can be sine squared of x over two. And then by the half-angle identity, this is the same thing as one-half times one minus cosine of x like so. And so we can start with that, all right? Which we can, again, write that as a fraction. Cause again, we're trying to write this as a fraction right here, so let's rewrite this as a fraction. You get one minus cosine of x over two, right? Let's compare to where we're at. We want a two tangent on the bottom. So it seems very tempting just to times the bottom by tangent of x. If we do that, we have to do that to the numerator as well. What happens if we distribute the tangent here? Well, you get one times tangent, which is a tangent. And then you're gonna get tangent times cosine. So let's investigate this. This seems to be a fruitful direction to go. You're gonna get tangent of x minus cosine of x times tangent of x like so over two tangent of x. Well, we want this product cosine times tangent to be a sine, otherwise we'd be done, right? So if we think of tangent in terms of sines and cosines, that would look like tangent, of course, we know is sine over cosine. And so Bob's your uncle right there. We see that there's a tangent. When we did the tangent, you got sine over cosine. There's a cosine now in its denominator, which cancels with this cosine. And so this would simplify to give us tangent of x minus sine of x over two times tangent of x. So we were able to prove this trigonometric identity here because this is the right hand side. We were able to prove this trigonometric identity using the half angle identity. And in particular, when you're using the half angle identity, the original version we have here with the plus or minus the square root is not always the easiest one to use. Oftentimes we prefer the square right here. So you get sine squared of x over two is equal to one half one minus cosine of x. Or an alternative version here as you get sine squared of x is equal to one half one minus cosine of two x. This is a very useful version. Likewise, cosine squared of x is equal to one half one plus cosine of two x. These versions of the half angle identities are gonna be probably much more useful for you when you're proving other identities themselves and other calculations as well. So be aware there are multiple versions of the half angle identity. And that brings us to the end of lecture 19 about the half angle identity. Thanks for watching everyone. If you've learned anything in these videos, feel free to give us a like, subscribe if you wanna see more videos like this in the future. And as always, if you have any questions, feel free to post them in the comments and I'll be glad to answer them for you. Bye everyone.