 In this video we present the solution to question number 13 for the practice exam number 2 for math 1060 In which case we're asked to prove the identity 2 cosine squared of theta over 2 equals sine squared theta over 1 minus cosine theta I always like to start with the more complicated side, which I'm gonna argue would be the left-hand side here It's easy to see the theta over 2 the half angle identities coming into play at some point I want to take care of that as soon as I can because after all The left-hand side has a theta over 2 the right-hand side only has theta So I have to some at some point in the identity convert from theta over 2 to theta And so the half angle identity is what I'm going to use there So the left-hand side is equal to 2 cosine squared of theta over 2 So recall that the half I think the half angle identity for cosine is the following cosine of theta over 2 is equal to plus or minus the square root of 1 plus cosine theta over 2 It's much more fruitful, especially in this case to use the square of the half angle So cosine squared of theta over 2 this equals just one plus cosine theta or I like to think of it as one half One plus cosine theta. So I'm actually gonna make this substitution in right here So we get by the half angle identity. We're gonna get two times one half one plus cosine theta Like so and then in which case then you see the two times the one half they cancel out And so then the left-hand side simplifies just to be one plus cosine theta now at this point Maybe you get a little it's stuck on what to do now what personally I would do at this point is I'm gonna look at the denominator and see that okay look where I'm trying to go towards I'm trying to go towards sine squared over one minus cosine theta that to me I see the one over my I see the one minus cosine the bottom I have a one plus cosine currently so this this to me personally is like oh I should multiply by the conjugate one minus cosine theta Over one minus cosine theta like so now if you didn't see that that's okay And another way of doing this might just be to take the right-hand side and start evaluating it You can do a little bit from the left a little bit from the right just never make just never do it at the same time That's what you have to watch out for and so then if you're looking at this one on the right-hand side You might think of multiplying by conjugates as well Maybe using some pipe of Pythagorean identity Help you out here, but I'm gonna stick with the left-hand side here if we foil out the numerator We're gonna get one minus cosine squared more specifically a one times one Which is one one times negative cosine cosine times one those cancel out and then you get a negative cosine squared This it's above of course one minus cosine theta. This is the denominator. We're looking for right here So that's that's a good sign. What about the numerator? Of course To get the numerator while one minus cosine squared. There's some quadratic terms there That makes me think about the Pythagorean relationship Notice of course that cosine squared theta plus sine squared theta is equal to one This tells me of course that sine squared is equal to one minus cosine squared Which is exactly what we're looking for isn't it we end up with then next sine squared theta over one minus cosine theta And that's equal to the right-hand side like so when working on these trigonometric identities It's very important that you always write down the angle on this question It's very critical because you have different angles in play You have a theta over two and a theta we should never write things like for example We should never write cosine squared plus sine squared equals one That's how we say it in words, but in terms of written notation our notation should be precise So we need to include the theta here if we don't mention the angles when we prove trigonometric identities We're gonna get some demerits on an exam So make sure you always write the angle out especially on this one where different angles are being involved in the same and the same proof