 Hello, welcome to another session on trigonometric problem solving. So we are going to take up one trigonometric Identity and we are going to prove that identity So the question says prove that sin theta upon cot theta plus cosec theta is equal to 2 plus sin theta cot sin theta upon cot theta minus cosec theta Now we are going to discuss two approaches here One approach is where we take the LHS and do manipulations and finally get the RHS Another one is we adjust the identity itself so that We reduce it to a more comfortable form So we'll see both the approaches. So let us see the first approach where we take the LHS and go to the RHS So in the LHS it is given sin theta upon cot theta plus cosec theta and if you notice on the RHS you have the same Trigonometric ratios, but then instead of a plus in the denominator, there is a minus sign So why not we just introduce this particular term in the LHS and then try to reduce the identity now So what what am I going to do is I'm going to multiply sin theta upon cot theta plus cosec theta given in the LHS by cot theta minus cosec and theta and We'll multiply and divide both the numerator and denominator by this So it's it's mentioned over here. Now. Why am I doing this? I'm doing this because I will get this sin theta and this cot theta minus cosec theta term Right, which is required here in the RHS, so I'll segregate that so that's what I have done So I've segregated that what was required and then now whatever was whatever is residue I'll try to reduce it to the desired result So this term is sin theta divided by cot theta minus cosec theta and now I am again multiplying and dividing by cot theta minus cosec theta Again, why? Because if because if you see we know the strict that if cot theta plus cosec theta is multiplied by cot theta minus cosec theta We'll end up getting this identity. So we'll be using which identity cosecant cosecant square theta minus cot square theta is 1 Right. So in this case, it's cot, cot, cot square theta minus cosec square theta, which will be minus 1 But then eventually we'll get a simplified form. So hence sin theta and this term was desired anyway So I'm not going to touch it On the RHS the denominator will be reduced Sorry, and the same thing in the right-hand term the denominator is going to be reduced to minus 1. Why? Because in this case Cot square is ahead of cos square that is cosec square theta is being subtracted from cot square But the identity is this where you are subtracting cot square from cosec square So hence it is minus 1 and in the numerator. I have just used another algebraic identity, which is a plus b Sorry a minus b it is so it is a minus b Whole square is a square plus b square minus twice a b So this is what I have used here And I've expanded Expanded the identity Now I expanded this term. So hence you'll get cot square theta plus cosec square theta Minus 2 cot theta cosec theta and divided by minus 1 which we just saw now So I'm not going to touch this part anyways because this is desired So what is left over on on the right term? If you see cosec cot square theta, I have just You know taken away this negative sign from the denominator and hence Reverse the signs of whatever was there in the numerator. So minus Minus 2 cot theta cosec theta has now become 2 cot theta cosec theta And cot square theta plus cosec square theta, which was here They are all now negative. Why because I have taken away the negative sign from the denominator now So again this term I'm not going to touch. Why because it's going to be there in the rhs So what is left now? So 2 cot theta into cosec theta Minus cot square theta and now I have reduced Cosecant square as cot square plus 1. Why because we know that cosecant square theta is equal to 1 plus cot square theta same identity which one This one if you rearrange it, you'll get the same thing So I have now reduced it to that that and then after simplification It is to why am I doing this because I am interested in this factor 2 which is there in the rhs So hence I reduce it like that And so after simplification. This is what is left in this term again. No, I'm not touching this So hence now I'm writing cos cot into cos by sine cos cot as cos by sine here And cosecant as 1 upon sine here and minus 2 cot square will be 2 cos square sine square theta and minus 1 Now hope you understood till this step now again this term. I'm not going to touch Now I take LCM in the right term sine square theta is the LCM. So it becomes 2 cos theta minus 2 cos square theta here and then since it was minus 1 so it will be minus sine square theta now Again, I have to get 2 somehow. So but there are two twos and one one sine square and I have to also get 2 plus 1 something like that. Why because you know, um, I have to get 2 into Some factor here and then I have to also get to 1 because Here if you see there is So the whatever is the multiplicand here. So if you see this Will be there. So hence in this term there has to be 1 so that this term is left over And since 2 is also there on the rhs so to multiply by this reciprocal reciprocal of this term must appear That's what the thought process is Let us proceed. So hence now Sine square theta can be written as minus 2 sine square theta plus sine square theta, isn't it minus sine square theta can be written as Minus 2 sine square theta plus sine square again. There is a purpose. Why? Because it can I can extract 2 from here So if you see 2 cos theta is there and then minus 2 common From cos square theta plus sine square theta will reduce it to 1 And then this sine square theta when divided by this sine square theta here is this So I have separated the terms now So sine theta by this term again is not touched now 2 cos theta minus cos square theta plus sine square theta is 1 That reduces to minus 2 because there was a minus 2 here also So 2 cos theta minus 2 plus 1 by sine square theta now I got this one because this one when multiplied by This will give you the one term of the rhs. What was that? Here this term is now in foresight now if you see Yes, so what is what is left now? So hence now I'm again reducing 2 cos theta. I'm taking 2 as common. So cos theta minus 1 Divide by sine theta right into 1 upon so sine square theta can be written as sine theta into sine theta This is all by purpose. I'll show you how now cos theta 2 cos 2 times cos theta by sine theta minus 1 I broke this Fraction into 2 fractions. So cos theta by sine theta minus 1 by sine theta here And then in the hole is divided into multiplied by 1 by sine theta and then plus 1 now This term again, let it be like that. Now this term is reduced to Cos by sine is cot. So here is the cot and 1 by sine is cosec theta and 1 by sine is as it is and plus 1 is as it is Okay. Now this again is as it is and then what will happen? Uh This 2 into cot theta minus cosec theta divided by sine theta this term right and then We have multiplied this whole term by the first term here And this whole term by this one Okay, so you'll get two terms This one and this one This one was desired On the rhs and if you see these this first term here when multiplied together you'll get 2 Why because this sine theta will go by this sine theta and this cot theta minus cosec theta will go by this cot theta minus cosec and theta So you get the desired result 2 plus sine theta cosec Sine theta divided by cot theta minus cosec theta Correct. Now, what was the learning learning is you can You can uh, you know have a you know a track of rhs. What exactly is needed in rhs You try to get that in lhs itself and then reduce whatever is the remaining term And simplify it so that it gives you the desired result. So that's what we did we Purposefully inserted cot theta minus cosec theta here under sine theta so that we get one of the terms And then kept on simplifying the other half of the expression so that it reduces reduces to the desired result Alternatively, you could have done this and this is much simpler method What you could have done is the same identity can be rewritten as like this right, so I'm just Taken sine theta upon cot theta minus cosec and theta which was on the right hand side I have taken it to a to the left hand side now simplify lhs So you take lcm you will get cot square theta minus cosec and square theta And then the remaining in the numerator will become sine theta times cot theta minus cosec and theta And minus sine theta times This cot theta Plus cosec theta that's what I've written now simplify You will get sine theta cot theta minus sine theta cosec and theta Minus sine theta cot theta minus sine theta cosec and theta and if you simplify again You'll get minus 2 times sine theta cosec and theta divided by minus 1 Why because this is again an identity which identity it is we have learned this That cosec and square theta minus Cot square theta is always 1 so hence it will be minus 1 because the signs are opposite So minus 1 so if you see It will be reduced to minus 2 so this thing goes Um sine theta into cot theta gets cancelled because of the sign So hence minus 2 times sine theta cosec and theta divided by minus 1 And we know that sine into sine theta into cosec and theta is 1 why because sine theta is 1 upon Cosic and theta so hence you get 2 lhs is a much simpler method than The previous one So this can also be adopted Thank you