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Edexcel A Level Mathematics

Pure Mathematics

Trigonometric Formulae

In this video, the speaker demonstrates how to prove that cot y equals sec x plus tan x, given that sine of y equals cos of x plus y. To begin, the speaker notes that the compound angle formula can be used because the cosine function has x and y in it. The speaker then uses the formula for cosine x plus y, substituting it into the equation to begin simplifying it. The speaker divides everything by sine y and brings out the cot y term. Then, they divide everything by cos x, bringing out the sec x and tan x terms. The final step involves rearranging the terms to achieve the desired equation. The speaker concludes that practice is key to mastering this type of problem.

Hello hello and welcome. In this video we're looking at a proof using the compound angle formulas. So in this question it says that given that sine of y equals cos of x plus y show that cot y equals sec x plus tan x. So when you first see this it might look like this stuff on the left has nothing to do with any of this, which is kind of right. This doesn't look anything like it. The only similarity really is that this is in terms of x's and y's and we have sine and cosine functions and this is in terms of x's and y's. And although it's not written in terms of sine and cosines it is in terms of cot sec and tan which can be rewritten as sine and cosine. So the key thing to notice here when you go to prove these is that here inside this cosine function we have an x and a y and that means that we can use the compound angle formula. So the first thing I would do is write out the formula for cosine x plus y which is this. Remember this is one of the formulas to be given in the formula book but you need to know how to use. And once we know this we can then substitute all of this into here. So basically the first thing I'm gonna do is I'm gonna write this out and I'm gonna write that sine of y equals and instead of writing cos of x plus y I can write all of this. So I'm gonna write cos of x cos of y minus sine of x sine of y. Yeah so I haven't really used any proper logic yet all I've done is use this formula to substitute this for all of this. And now I can start making this look like this stuff which is a bit tricky. One thing to notice here is that we have cots and tans and stuff whereas here we have loads of sine y's. So what I might do is divide everything by sine y because if we divided this, well I write it all out, we divide this by sine y that'll become one in a minute. We have cos x cos y over sine y. This cos y over sine y part could become a cot y and we have minus sine x sine y over sine y which will cancel out this sine y. So overall now we have 1 plus cos x cot y minus, and these two cancel out, so minus sine x. Yeah so this now isn't quite looking like what we want. At least we can see the cot. Yeah we can see the cot y and now I do the same thing but I can divide by cos of x because that will cancel out this cos of x and I'll turn this into a tan. So 1 divided by cos x is sec x and then cos x cot y divided by cos x is going to cancel out the cos x so I'm going to get cot y and sine x divided by cos x is tan x. And now hopefully you can see that this here is very similar to what we have up here. All I really need to do is manipulate it, we'll switch around a little bit. So I'm going to move my tan x to the left along with my sec x and I'm going to leave my cot y on the right and that now is exactly the same thing as what we wanted. I mean I know mine's switched around but that doesn't matter we've we've proven it, we've shown it. And that's it. So really this isn't the easiest thing to do but the key point is to notice that here we have x plus y so we have to use one of our compound angle formulas and then you're just trying to get this really horrible looking thing into something much more simple. And because we have so many sine y's and cos x's we're going to be dividing by those to turn things into tan and cot. And that's it. So hopefully that all makes sense. The deal with these things is you really need to practice a fair few and the more you do the more proofs you try the easier it gets. You start to kind of see these things a bit more. So anyway hope that was helpful and I'll see you next video. Thanks.

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