 okay good morning guys write your names so that I can know who are attending the class Ritu is there Shraddha is there Tripan this is for class 9th okay Ancita is there Tana here Varsha Sanjana six people are here I can see eight people said eight eight eight people attending okay let me start the class can anyone post the questions on off trigonometry that we left on that day can anyone take the pic and send it to me on my WhatsApp so that I can start from there and if I get time I'll move into circles okay thank you Shraddha for posting it let me solve this question for you I'm writing down a question so you have to take left hand side this is your LHS and you have to modify LHS you have to work out on LHS to find out RHS so don't look at the RHS now the RHS sorry look at the RHS now the RHS is in terms of sec theta and cosic theta so sec theta is a form of cos theta and cosic theta is a form of sin theta it means that whatever values have been given here in terms of tan theta and cos theta should be converted into sin theta and cos theta in such a manner that you get this particular value in the right hand side so start solving this question are you guys solving it okay Dhoti welcome to the class should I solve it anyone want one more minute okay Ancita I'll explain you don't worry I'll solve you Ancita for you don't worry okay let me solve it for you so those who are those who were not present in the class just learn each formula one by one tan theta can be written as sin theta by cos theta and cot theta which is one by tan theta can be written as cos theta divided by sin theta so what I'm trying to do is that I'm trying to replace tan theta by sin theta by cos theta divided by one minus cos theta by sin theta so let me solve it first this is nothing but sin theta by cos theta complete thing divided by let me cross multiply it it becomes sin theta minus cos theta divided by sin theta so this is nothing but sin theta if I write it in terms of division sin form sin theta minus cos theta divided by sin theta so the sin theta goes in the numerator it gives me sin square theta divided by cos theta sin theta minus cos theta so first thing comes out like this so now let me go to the second one and let me make some space for you guys so cot theta can be written as cos theta divided by sin theta whole thing divided by one minus cos theta divided by sin theta so similarly the sin theta one second I have written it opposite so the whole thing can be written as one minus sin theta divided by cos theta so as I have done here this cos theta will go in the numerators so this becomes cos square theta sin theta cos theta minus sin theta so guys write it down till here because I'll change the screen just write it down till here I'm giving you 30 seconds just write it down till here okay I hope you have done it now what I have to do is that I have to add these two things sin square theta divided by cos theta sin theta minus cos theta plus cos square theta divided by sin theta cos theta minus sin theta now what I'll do see here it is sin theta minus cos theta here it is cos theta minus sin theta so I'll take an I'll take a negative outside and this becomes sin square theta divided by cos theta sin theta minus cos theta minus cos square theta divided by sin theta sin theta minus cos theta so now what happens when you take LCM this LCM comes out to be sin theta cos theta sin theta minus cos theta now this cos theta cos theta and sin theta minus cos theta will get cancelled out this will become sin cube theta minus here sin theta and sin theta minus cos theta comes out to cos theta into cos square theta is cos cube theta so how much is left out look at here this can be written as sin cube theta a cube minus b cube can be written as a minus b a minus b a square plus b square plus a b divided by sin theta cos theta sin theta minus cos theta now this sin theta minus cos theta will get cancelled out here this sin square theta plus cos square theta I can write it down as 1 so it comes out to be or let me write here this is after this step I'm writing it over here is it okay so this comes out to be 1 plus sin theta cos theta divided by sin theta cos theta so when you divide this 1 by sin theta cos theta this changes to sin theta cos theta and when you divide sin theta by cos theta by sin theta cos theta this changes to 1 so this comes out to be 1 plus sin theta cos theta which was to be proved now if anyone did not understand you can write it in the chat box I do it for you please write it in the chat box I'll do it for you okay nobody is saying me anything so I'll give you another question which is not from your book the question is same kind of question same kind of question cos theta divided by 1 minus tan theta plus sin square theta divided by sin theta minus cos theta would be what I'm not giving you RHS I'm just saying simplify can you solve this question for me should I start solving this is very easy question anyone needs more time nobody is saying anything should I start solving okay let me solve so everything is in form of cos theta plus sin theta somebody is saying let me check everything is in form of sin and cos except this tan so I already said that tan theta is equal to sin theta divided by cos theta so I'll change it in that form this becomes cos theta 1 minus sin theta divided by cos theta to do it more this cos theta I'll take it here and from denominator it will go to the numerator so this becomes cos square theta cos theta minus sin theta which is equal to minus cos square theta sin theta minus cos theta so here you have sin square theta so now I'm writing it in this format sin square theta divided by sin theta minus cos theta minus cos square theta divided by sin theta minus cos theta which is equal to sin square theta minus cos square theta divided by sin theta minus cos theta so let it separate this becomes sin theta plus cos theta sin theta minus cos theta divided by sin theta minus cos theta so this and this gets cancelled out the answer comes out to be sin theta plus cos theta if anyone has any doubt you can ask me you can post it in the chat box quickly if you don't have reverse the order means Shraddha this is the way you have to solve that is fine if you have written cos theta plus sin theta that's equal to sin theta 2 plus 3 is equal to 3 plus 2 that's that's not a wrong thing to do okay fine now let me do another question for you okay this is a good question second theta plus tan theta is equal to x then find x square minus 1 divided by x square plus 1 solve this question have you done it there's a free delivery okay few people are getting a square few people are getting something else now let me solve this first let me tell you this when I told you this formula let me discuss formula and those who did not those who are not present please look at this I told that sin square theta plus cos square theta is equal to 1 you divide the complete equation by cos square theta so I do it like this sin square theta divided by cos square theta plus cos square theta divided by cos square theta plus 1 by cos square theta so how much it is this is tan square theta plus 1 is equal to sec square theta so this formula is going to be used in this in this particular example or sorry in this particular question and this particular formula is going to be used so as this has been given x square minus 1 and x square plus 1 first I will find out x square so x square would be sec square theta plus tan square theta plus 2 sec theta tan theta so what do I get here I have two things to do so if negative sign has been given here so I am writing x square minus 1 which is x square theta plus tan square theta plus 2 sec theta tan theta and it is divided by x square plus 1 so I am writing sec square theta plus tan square theta plus I do minus 1 here I left minus 1 here plus 2 sec theta tan theta plus 1 so look at here sec square theta minus 1 would be if you take minus 1 here sec square theta minus 1 would be tan square theta so what I am writing over here in the numerator is this this becomes tan square theta you already have tan square theta and this becomes 2 sec theta tan theta now this sec square theta sorry this tan square theta plus 1 becomes sec square theta so you already have sec square theta here and tan square theta plus 1 will become sec square theta and you have 2 sec theta tan theta so what comes out is that you look at here this comes out to be 2 tan square theta plus 2 sec theta tan theta divided by 2 sec square theta 2 sec theta tan theta so what I will do is that let me go to the another page this is the thing 2 tan square theta 2 sec theta tan theta divided by 2 sec square theta 2 sec theta tan theta so what I will do is that from here I will take out 2 tan theta sec theta 2 tan theta out so this comes out to be 2 tan theta so what is left out tan theta plus sec theta is left out from here I will take out 2 sec theta so 2 sec theta if I take out sec theta plus tan theta is left out now this sec theta and tan theta will get cancelled out so what will happen is that and this 2 and 2 gets cancelled out tan theta divided by sec theta or I can write it 1 tan theta 1 by sec theta is equal to cos theta so 1 by sec theta here I am writing as cos theta so tan theta is equal to sin theta divided by cos theta into cos theta so this cos theta cos theta gets cancelled out you get sin theta as an answer so did you guys understand it should I explain it once more should I explain it once more the last part Sraddha is saying so what I am saying is that 2 is common so I am taking 2 out here tan theta is common so I am taking tan theta out here so if you take 2 tan theta out here tan theta will be left out and here sec theta will be left out so sec theta is left out here now here 2 sec theta is common so I am taking 2 sec theta out so this is sec theta left out here and tan theta left out here tan theta plus sec theta divided by sec theta plus tan theta they are equal I have cancelled it out now 2 2 gets cancelled out it is left out tan theta divided by sec theta now 1 this is nothing but this is nothing but tan theta multiplied by 1 by sec theta 1 by sec theta is cos theta so I have written it tan theta into cos theta tan theta into cos theta changes tan theta changes to sin theta divided by cos theta and cos theta and cos theta is cancelled out so we are left out with sin theta so that is how it has to be done Sraddha did you understand now okay write down another question I am taking it from your book so I am doing this question which is 1 plus sec a divided by sec a prove that would be equal to sin square a divided by 1 minus cos a solve this question and look at the hint I have taken this question from your book look at the hint the hint has been given as simplify LHS and RHS separately it means that only doing LHS will not give you the answer you have to work on RHS also so once you find out answer of LHS start working out on RHS so that both the answers can be proved equal could I solve it good Dhruti has got the answer Ritu and Sraddha has got the answer good so 1 by sec a plus sec a by sec a so 1 by sec a is cos a sec a by sec a is 1 so left hand side is 1 plus cos a now 1 sin square a plus cos square a would be equal to 1 so sin square a can be converted into 1 minus cos square a now what is the logic I want everything in terms of cos because my left hand side is coming in terms of cos that is why sin has been converted to cos like this so sin square a can be written as 1 minus cos square a divided by 1 minus cos a so this can be written as 1 minus cos a 1 plus cos a divided by 1 minus cos a now this 1 minus cos a 1 minus cos a will get cancelled out and RHS will also come out to be 1 plus cos a so that's your answer so a lot of people have got this answer so let's solve another question this question is next question cos a minus sin a plus 1 divided by cos a plus sin a minus 1 is equal to cos a plus cot a so take our LHS prove RHS start solving guys give me a moment I'll be just back in one minute hello have you done it what is the answer I got two people giving the answer how do you do it so try to understand and this is cos a minus sin a plus 1 divided by cos a plus sin a minus 1 so multiply it with rationalize it so take complimentary of it cos a minus sin a minus 1 and this will become sorry cos a plus sin a plus 1 and this will become cos a plus sin a plus 1 so this has to be multiplied so this is nothing but look at here this this I am just rearranging in some form so numerator can be written as 1 plus cos a minus sin a and this numerator can be written as 1 plus cos a plus sin a divided by this numerator this can be written as cos a plus sin a whole square because if this is a this is a and this is a minus b and a plus b form so a square minus b square form and this also can be written as a square a minus b a plus b so a square minus b square form so this can be written as 1 plus cos a square minus sin square a divided by this can be written as cos square a plus sin square a plus 2 sin a cos a minus 1 so I am again solving it this can be written as 1 plus cos square a plus 2 cos a minus sin square a and if you look at this very properly this cos square a plus sin square a would be 1 so this one and this one will get cancelled out so in denominator only 2 sin a cos a would be left out now this 1 minus sin square a can be written as 1 minus sin square a can be written as cos square a so I can write this as 1 minus sin square a is cos square a and I have this extra cos square a so I can write this as 2 cos square a plus 2 cos a divided by 2 sin a cos a here you have a guys so take 2 cos a common you are left out with cos a plus 1 divided by 2 sin a cos a so cos a cos a 2 2 gone this comes out to be cos a plus 1 divided by sin a so cos a by sin a is equal to cot a 1 by sin a is equal to cos a so this is your answer so that's how you have to do this question this was this is supposed to be a difficult question at class ninth level but that's how it has to be done now anybody has any doubt you can ask me any doubt you can type it in the chat box okay I'm still waiting okay I'm not getting any anyone asking me any doubt so I'm giving you another question from your book only solve this question 1 plus tan square a divided by 1 plus cot square a is equal to tan square a the solve it done okay great so what I write over here is that 1 plus sin square a divided by cos square a divided by 1 plus cos square a divided by sin square a so this can be written as cos square a plus sin square a divided by cos square a and this can be written as sin square a plus cos square a divided by sin square a so this cos square is science sin square a any well it is once it will get cancelled out so it comes out to be 1 by cos square a divided by 1 by sin square a so sin square a goes in the numerator divided by cos square a and that gives me a value of 1 okay let me solve next question sin a plus cosic a plus cos a plus sin square you have to prove that this is equal to 7 plus tan square a plus okay great one person has done it I'm waiting for others Dhruti also has done it good so Dhruti and Ritu has done it I'm waiting for others okay Varsha has done it Shraddha has done it so most of you have done it let me solve this question so first what I do is that I'll simply multiply sorry square it so this comes out to be sin square a plus cosic square a plus 2 sin a cosic a now the sin a cos a k will get cancelled out so this will become 2 plus sin square a plus cos x square a now this cos x square a I'll convert it to sin square a so this becomes 2 plus sin square a plus 1 by sin square a so this is nothing but sin to the power 4 a plus 2 sin square a plus 1 divided by sin square a similarly this can be written as cos square a plus sin square a plus 2 cos a sin a so this is nothing but cos square a this cos a sin k will get cancelled out so cos square a plus sin square a plus 2 so which is nothing but cos square a plus 1 by cos square a plus 2 so which is cos to the power 4 a plus 2 cos square a plus 1 divided by cos square a now how do you solve this so to solve it I have to add it so if you look at it this is nothing but sin to the power 4 a plus 2 sin square a plus 1 divided by sin square a and this is nothing but cos to the power 4 a plus 2 cos square a plus 1 divided by cos square a so it can be written as this is nothing but 1 plus sin square a whole square let me check it once so this can be written as 1 plus sin square a divided by sin a whole square this becomes actually give me a moment so we will not do it like this what we will do is that we will simply multiply it with we will simply multiply this with cos square a so this becomes cos square a sin to the power 4 a 2 sin square a cos square a plus cos square a and this is nothing but sin square a cos to the power 4 a plus 2 sin square a cos square a plus sin square a and this whole divided by so let me solve the numerator I will come to the denominator later I am just writing here for this equation denominator d is sin square a cos square a so you look at two terms first cos square a plus sin square a will give me one now this is 2 sin square a here and 2 sin square cos square a here and 2 sin square cos square a here so this will give me 4 sin square cos square and I have cos square a sin to the power 4 a cos to the power 4 a sin square a so I have taken all terms in the consideration this and this are done this and this has been written as 1 and this 2 plus 2 has been written as 4 so now what happens is that 1 plus 4 sin square a cos square a and here you take sin square cos square common so you take sin square a and cos square a common you are left out with sin square a plus cos square a which is nothing but this is nothing but 1 so this comes out to be 1 plus 5 sin square a cos square a and because this is sin square a cos square a this will get added up so what we had to prove we had to prove that and when you divide it by sin square a cos square a this comes out to be 5 plus 6 square a cos square a now let me take the right hand term so right hand term over here is 7 plus tan square a plus cot square a so 7 plus sin square a divided by cos square a and cos square a divided by sin square a so this would be 7 7 sin square cos square plus sin to the power 4 a plus cos to the power 4 a divided by sin square cos square now if I take 2 sin square cos square a from here this becomes 5 sin square a cos square a plus sin square a plus sin to the power 4 a plus cos to the power 4 a plus 2 sin square cos square sin square a cos square a so this is to the power 4 to the power 4 and this is 2 I have taken 7 bifurcated into 5 and 2 divided by sin square a cos square a I'm writing it here this can be written as 5 sin square a cos square a and this is nothing but sin square a plus cos square a whole square divided by sin square a cos square a so this is one so I can write it 5 plus 1 divided by so this becomes second square a cos x square a so LHS becomes equal to RHS so that's how you have to solve this question this was a difficult question okay so what I did Shraddha was I didn't simplify it in the beginning perhaps I might have lost a step here what you can do is that here when you are writing 5 sin square a cos square a you can convert it into sin to the power 4 a plus this one can be written as sin square a plus cos square a whole square what you can do over here is as simple as that and then you can convert it one way to solve this question is I am solving from here just look at here I'm making some space one other way to solve this question is you simplify it from here only this this was once 5 sin square a cos square a so this can be written this one can be written as sin square a plus cos square a whole square this is one plus 5 sin square a cos square a divided by sin square a cos square a so just open the bracket you will find out sin to the power 4 a plus cos to the power 4 a plus 2 sin square a cos square a plus 5 sin square a cos square a divided by sin square a cos square a so I am writing it here this can be written as sin to the power 4 a plus cos to the power 4 a plus 7 sin square a cos square a divided by sin square a and cos square a now you just have to simply divide it you will get your direct answer so how do you divide it I'll tell you now so to divide it what I can do is that this sin to the power 4 a divided by sin square a cos square a will convert to sin square a divided by cos square a this cos to the power 4 a divided by cos square a would be cos square a divided by sin square a plus 7 so this can be written as 7 plus tan square a plus cot square a it can be done like this also yes Radha I can understand you can do it I I missed that step so I'm giving you two methods third method by your method I'll do it in the class because time is already done so solve the other questions and let me know if you have any doubt do you have any doubt any doubt anyone okay I'm not getting any response from you guys so thank you so much for joining the class we'll meet you next time whether online or on the or in the class NPS R&R let us check what's the thing so thank you so much okay