 Hello friends, so we have another question on proving the trigonometric identity and this question is 1 plus cot theta plus tan theta times sine theta minus cos theta is equal to secant square secant theta by cosecant square theta minus cosecant theta by secant square theta Now again, so first thing to be understood here is what should be the approach to solve such problems Okay, so if you can see on a first glance on the right-hand side if you can see it's nothing but 1 upon cosecant square is sine square Is it and secant theta this secant theta is 1 upon cos theta so in a way this RHS is nothing but sine square theta by cos theta minus cos square theta by sine theta Right and if you look at the LHS you will definitely get sine theta and cos theta in the denominator because there is cot and tan So hence the approach is to reduce cot and tan into sine into sine and cos terms So hence when I write that though I can write 1 plus cot can be written as cos upon sine and Tan can be written as sine upon cos is it and the right fact the factor which is there in the second Position that is sine theta minus cos theta. This is that factor now What do I do is I take LCM in the second in the first factor. So hence it is sine theta cos theta is the LCM Then the first when you when multiplied by 1 the first term is sine theta times cos theta Then since sine theta was already there. So cos will become cos square and this will become sine square The third term is sine square times sine theta minus cos theta which was there Now we know we know one identity in basic identity is sine square theta Plus cos square theta is one So if you look at here this particular term gets reduced to one and that one is written over here in the next step Now that one plus sine theta plus cos theta sine theta times cos theta Which was this term is written here sine theta cos theta Divide by this sine theta cos theta is written over here sine theta cos theta right and now we open up the bracket Okay, so we multiply The numerators so if you multiply the numerators you'll get sine theta minus cos theta Plus sine square cos theta minus sine theta cos square theta you can do that and divide by sine theta cos theta is there in the denominator Now once you open this these two factors now, what is the what is that catch catches? Now don't club the alternate terms that is whatever you multiply you don't club them again Otherwise you'll get you'll go back to step the previous step right instead of that what I'm now going to do is I am clubbing this sine theta with this this term Why because I can take sine common and you can see then you will be getting 1 minus cos square Which will be equal to sine square. So I'm moving towards the target Similarly here also if you can take cos common there is 1 minus sine square term. So which is Going to be you know reducing our Or we are moving towards our target So hence I take sine theta common. So sine theta 1 and times 1 minus cos square theta Minus cos theta times 1 minus sine square theta So hence what happens this 1 minus cos square theta is now sine square theta and this 1 minus sine square theta is cos square theta Right by which identity this identity only now. This is nothing but sine theta times sine square I'm purposefully not writing sine cube because you will see now that will separate this fraction So I'm purposefully writing sine cube as sine theta into sine square divide by sine cos, right? And similarly the second term now if you see this sign and this sign will go So hence you will be reduced it will be reduced to sine square upon cos and This cos and this cos will go so it is reduced to cos square by sine. This is what we intended to find Can you see this we got the result right and then you can just Write in their reciprocal form. So this is secant theta upon cosecant square theta Minus cosecant theta divided by secant square theta, which is the desired RHS So hence we could prove this identity. So what was the learning you have to take care of both? LHS and RHS keep track of both try to see from where you are starting and from where and to what point Do you want to go to so this particular thing was you know This reducing this into sine and cos helped us and hence we could attain the result