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Sin & coSq e sin & singez + cos & e e: = @1 = COS0 cosqe1 cos & Fsing e_ sin & e: ec =e1 = sin ( e + cosq ez...

Question

Sin & coSq e sin & singez + cos & e e: = @1 = COS0 cosqe1 cos & Fsing e_ sin & e: ec =e1 = sin ( e + cosq ez

sin & coSq e sin & singez + cos & e e: = @1 = COS0 cosqe1 cos & Fsing e_ sin & e: ec =e1 = sin ( e + cosq ez



Answers

$$ \frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}=2 \operatorname{cosec} A $$

So here we need to prove that our alleges is equal to our marriages. Okay Our alleges is one plus card a place turn a into sign of a minus cause of pain. And our marriages here is set a by because ex choir a minus Kazakh vibe. Sex square eight. Okay so as usual we will again start with our NHS here. Okay you can start between the side. It does not make a difference. So let's begin. Like I always say first convert all of your known terms in terms of sine and costs. Okay they are our friendly towns. So cause by sign of a class sign of a by cause of eight in two years we have signed a minus cause a no taking calcium in the first record you always have sign it into cost. So you already have signed a less cause a okay and then the denominator will have you uh yeah I was a sine squared decor squaring and yeah we would have signed A. R. A. Okay. Yeah, we have signed a minus because of it. So now you open this cracker to make it a little bigger. So this will look like sign a minus cause of into. We have signed square. Let's go square a less sign a course and then sold by the advice to cause of the Okay. Now, if you look at this carefully, the numerator is a Q minus V cube which is equal to a minus B squared plus b squared plus 80. All right. So this will look something like this say cube minus house cube of A. There are advice signed a crossing no separating the numerator and denominator. Yeah, we have signed cube by plus minus here. We have prosecuted. You know, we have sunny egg toss it. All right. So this will give me Cancellelling one sign and then sign here and one cause and one cause here. Yeah. So you're remains to your also remains to now, what is the universe? Of course, this is a What is the inverse of science or, you know, we have. Does that square party than words of sign here? This will be closed. And there was, of course, will be six square which is equal to our marriages. Hence, through if you have any further doubts, please feel free to let me know. I'll be very happy to help. Thank you.

Now here again we have to prove that our LHs is equal to our rhs. Okay, what is our NHS here? Our NHS is signed A plus because A and to 1- sign a Because. All right. And our bridges here is sine cubed. A. Class cause cube. So how do we solve this? Uh, if you recall Okay, regulator solution first. All right solution. Now if you recall there were many asked african identities. Okay. Like A plus B hole square which was a squared plus B squared plus to a cave. Similarly, there was a traumatic identity which was sorry, an algebra entity which was a Q plus we cube. This was equal to a plus B into a square plus B squared minus. Amy. All right. Now, if you look at this carefully, this is our left hand side and this is our right hand side. This looks like a cue. This looks like beat you and this looks like a plus B. And this year looks like the other bracket which reaches this one. Okay so here we should we need to have here is square plus B square and we already have minus. Okay so now we need to find what is a square and be square in town in place of this one. Okay, so let's do that here. So I just use a different colour. So just to make it more colorful. Okay, so this is a sign it and then here we have for self care. Okay so now here we have 1- sign a. And then here we have parts of it. Now we know about the econometric identity which said that science square. The spots where it was 21. Okay, so if we replace this one with this Science credit score square, what do we have here? Let's see here we have signed A plus. Also if bracket open and that's right here. Sign square A. And then plus cause square in place of one. Okay. And then we are we right minus sign a Carson. Now look at this carefully. This shot exactly looks like A plus B into a square plus bees for r minus A B. Which is equal to Q. Plus b Q. Okay, so this is the formula. Alright, so from this, since you have obtained this here. Okay, so just right this way sign cube, A class boss cube. And when we have our marriage is okay. Hence grew. Now the tape here is that I'll just give you one small tape here. Okay. That tape here is that we don't always have to solve our equations with various calculations and up and down left and right just to reach from the left hand side of the right hand side. Okay. Sometimes what happens is that when you look at a question, the answer is looking at you dead in the eye. All right. When that happens, just look at the left hand side and I can tread carefully and you will simply easily know what you're supposed to like look at this. We didn't even do any calculations. We just added one extra time here in place of 10 science query plus cause query and then in just two simple steps we have our solution here sine cubed plus cost cuba which is equal to or marriages. Okay? If you have any further doubts, please feel free to let me know. I'll be very happy to help. Thank you.

So the question here is to again prove that LHs is equal to our NHS Alright, what is our alleges? Our alleges is one minus Kosek minus card A minus one by sino que. and our oranges here is one by sign of 8 -1 by concept. A scarred of Okay so while solving this problem we will come across a special condition. Okay without alleges will not be exactly equal to our marriage is okay in that situation what we have to do, we'll see in this problem. Okay this is a special case. It doesn't occur very often but if it does then this is the process we need to follow. Okay let's see one by Cossack minus court of minus one life sign. Ok so let's convert all of these in terms of silent costs. So this will be equal to one by sign a miner's because of a You know we have cause of any right Sign up here and then here we have one bite sign. Ok so first taking calcium over here we will have one bite 1- asked me by Sign A -1. Life. Scientific. This is equality sign here by one like this cost minus one by saying okay so taking L cm again here we will have sine squared a minus one minus cause a bye Sign into 1- because of it. Okay now converting sign square into one minus car squaring so minus here or then Yeah we have one minus parts of the right right Signed and until we have 1 -3. Yeah so now this is a square minus b square. This formula here this is a square minus B squared. So we'll write this as one minus cost into one place cause of a. Okay and you know we have minus one minus toss off the and then do I get right Some here 1- passed a So this is equal to taking common 1- costs outside. So here we will have one less cause of a -1 and until we have signed in and we have one minus boss 1- Cosine Cancell out. Okay and you have plus one and -1 cancels out. So here we have cause the right sign of this is equal to court. Now you can see this is not equal to our NHS isn't it? Look at this, this is something else. So what we'll do is that we hear we started to solve with our alleges in hand right now what we'll do is that we'll start to solve our RHS as well. Okay. No we'll start to swallow it just as well to see what comes. So here we have our our HSS one by signing a minus one by Kazakh airplane part of Okay so again changing this, I'll do this quickly. So here we will have one by sign of a plus. Here we have cause of a five sign up and then here we will have one by sign of a minus. We would have sign of a body one class cause of a taking calcium here what we'll have here one plus course of a Okay and then you will have minus sine square and then we will have to sign off into one place because of it. Okay so this will again week we'll do one plus or software and then you will have 1- costs square a the right right Signed eight times 1 less cost eight. So this is equal to one place cost a common and here we'll have one one minus cost. Alright so again here we will have sign off and then here we will have one place cost one plus cosine and plus cause it cancels out, plus one minus one cancels out. So here we will have minus minus plus that is equal to cause A and then you will have signed. This is equal to got it. Now, if you look at this carefully, our LHs is caught and our rhs here is also equal to caught. Okay this let's just is equal to our chest. Hence through. Okay, if you have any further doubts, please feel free to let me know. I'll be very happy to help. Thank you.

Okay. So in this question we are given an expression which we need to prove. We need to prove. That goes to the bar six X plus sign to the bar six X. Is equal to one minus three signs square eight and 2 Co Square. Okay, so beacons hit the power six and it's not easy to solve the power six. So we'll just Blow it down. Okay? We'll just bring it to the power of two or 3. So we know that A Q plus BQ is equals two A plus B. And to a square plus B square minus 80. Okay. And we also know that A plus B hole square is equal to is where let's be squared Plus two. Ap. Okay, I can see this similar time here. So I'll just replace this town with this. So what I get is a Q. Let's get you is equals two A plus B. And to a place behold square -3 of you. Okay, this is what I get now. I will use this expression I'll use this expression on the left hand side looking to solid. Now we can right Goes to the past 66 Plus sign to the bar 66 as co square whole cube. Sorry. Here we have to like exco square X. Hold you plus sine squared republicans. Okay. Why I have done this because I know that hair, I will arrive at something A plus B. Okay. And we know that square X plus science where X becomes one. So I have to break it down like this. So if I put uh this thing in this formula what I get is Goes to the past six x Plus sign to the past 66 is equal to sign square A science politics. Let's go square X. And two sign square X. Let's course medics. Phone sweat minus of three same squid X. And don't post connects. Okay, So now sliding. So now it's square. Now you can just see that cause to the past 66 plus sign to the bar. Sex sex Equals to this becomes one And this also becomes one and this is remaining saying, okay, Science Square it's and to forswear X. So this is our requirements, which we need to little which we had to prove. So hands proof. Thank you.


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