 Hello students, welcome to setup Academy YouTube channel. So where we bring for you every day a Question in out of problem solving which is extremely extremely helpful for the J main in advance aspirin's So guys and girls here we are again with you with one of the questions which has been picked up form binomial theorem chapter Okay, so now looking at this question. Now you realize that there is a series whose value is required So if you read the question, it says the value of Some ugly terms are there. Okay, and this is what basically intimidates people, right? So the question looks so ugly that half the people will think that they will not be able to solve this question and they'll move on But just wait a minute. This is why we exist here. We exist here to make your life simple So let's understand how do we solve this question? So guys and girls here we see few terms which have been constantly being repeated The first one which I like you to look at is e to the power i pi by 2 e to the power i pi by 2 e to the power i pi by 2 While this term is a very very simple term, which is basically written in a Euler's form So this term is actually 1 cos pi by 2 plus i sin pi by 2 in plain and simple word This expression is just an Euler form representation for i. Yes e to the power i pi by 2 whenever you see this anywhere in the question. It just stands for i iota nothing else Right next term that we see here is a root 2 plus 1 by 2 to guys and girls Are you familiar with this particular ratio? Have you ever seen this ratio while studying trigonometry? I mean even if you have you don't remember it, right? Because most of us remember the formulas from left to right We don't remember the formulas from right to left, right? So when you know the result you don't know whose result is it actually Right, so this expression actually is nothing but cos pi by 8. Yes This expression is actually cos pi by 8. How? Let's check it out. See we already know 2 cos square a Minus 1 is cos of 2a our famous double angle identity of cos isn't it? So if here you plug in the a value as pi by 8 you end up realizing that It'll become 2 cos square pi by 8 minus 1 equal to cos of pi by 4 Right and what's cos of pi by 4 cos of pi by 4 is 1 by root 2. Yes 1 by root 2 So 2 cos square pi by 8 is Equal to 1 plus 1 by root 2 which happens to be root 2 plus 1 by root 2. Isn't it? So this expression finally Gives us cos square pi by 8 as root 2 plus 1 by 2 root 2 Which means cos pi by 8 is under root of root 2 plus 1 by 2 root 2 Please do not put a minus sign because we know pi by 8 pi by 8 is 22 and a half degree It's in the first quadrant and the first quadrant cost is positive All right in the same way do we have do we have any idea about this term? So we have all the address this guy Now do we have idea about root 2 minus 1 by 2 root 2 now? I know there's no guess in that you can easily say that it stands for sign pi by 8. Yes So to root 2 minus 1 by 2 root 2 under root is basically nothing but sign of pi by 8 You can always try it out yourself using the formula 1 minus 2 Sign square a is cos of 2 a okay So please use this formula and figure out how sign pi by 8 comes out to be under root of root 2 minus 1 by 2 root 2 Right. All right now having said so What is this series all about? What are the series all about the series is basically nothing but an expansion of a binomial term? Isn't it? It's an expansion of a binomial term, right? And what is that binomial term? Let's look at it So let us call now. Let us call now a as root 2 plus 1 by 2 root 2 under root And let's call be as under root of root 2 minus 1 by 2 root 2 Right and and you already know that a is cos of pi by 8 and b is sign of pi by 8 But we'll come back to that a little later on So if you see this expression this expression actually is a to the power 64 then 64 c1 I Then you have a to the power 63 b to the power 1 then 64 c2 I square a to the power 62 b to the power of 2 and so on and so forth Which means you have actually written something of this type So the entire sum that I'm talking about the entire sum that you have is a to the power 64 Okay, you can back it up with 64 c0 doesn't hurt us to put that term is 64 c1 a to the power 63 I b whole raised to the power 1 right and the next one is 64 c2 a to the power 62 I b to the power 2 and so on and so forth Till we reach 64 c 64 a to the power 0 I b to the power 64 Now guys and girls this sum is clearly this sum is clearly the binomial expansion of a plus I b to the power of 64 right that is something which is not hidden from anybody if anybody knows the basics of Binomial expansion he would realize that this given sum this given series stands for the binomial expansion of this particular term Now it is here that we are going to substitute our a as course of pi by 8 and b as sine pi by 8, right? So you have cos pi by 8 plus I sine pi by 8 whole raised to the power 64 So whenever you see such a term what comes in your mind? Right the famous D more raise theorem the famous D more raise theorem So by the marvelous theorem we can multiply this power of 64 with these Arguments isn't it? So let's do that. So let's multiply 64 with these arguments and when we do that we end up getting 64 times pi by 8 So cos of 64 times pi by 8 plus I sine 64 times pi by 8 Which clearly simplifies to cos of 8 pi plus I sine 8 pi Cos 8 pi is clearly a 1 but sine 8 pi will be a 0 that leaves you with the final answer to be 1 So let's see which of the following options given to us Matches our results. Of course option number a is the right choice as you can see here Option number a is definitely definitely right. Thank you so much for watching. Stay safe. Stay healthy