 Hi, and how are you all today? I am Priyanka and let us discuss the following questions. It says proof that 2 cos Pi by 13 cos 9 Pi by 13 plus cos 3 Pi by 13 plus cos 5 Pi by 13 is equal to 0. Now in order to prove this first of all, we should be well versed with an identity. It says that 2 cos theta cos 5 is equal to cos theta plus 5 plus cos theta minus 5. We will be making use of this identity and hence it becomes a key idea before starting off with the proof. So let us start. Now we will be starting with the left-hand side of the equation that is given to us. Now here 2 cos Pi by 13 cos 9 Pi by 13 plus cos 3 Pi by 13 plus cos 5 Pi by 13 has to be reduced to 0. Now if you carefully analyze, our first term is in the form of 2 cos theta cos Pi because our theta and pi are there. So we can change them into cos theta plus pi plus cos theta minus pi and then these two terms will be written along with it. Cos 3 Pi by 13 plus cos 5 Pi by 13. Now on simplifying we have cos this will become 10 Pi by 13 plus cos. Now this will become minus 8 Pi by 13 plus cos 3 Pi by 13 plus cos 5 Pi by 13. Right? Now in order to simplify it further we can write cos 10 Pi by 13 plus cos 8 Pi by 13 plus now cos 3 Pi by 13 can be written in a different way that will be Pi minus 10 Pi by 13. Right? Because on simplifying this we'll get 3 Pi by 13 itself. Similarly cos 5 Pi by 13 can be reduced as i minus 8 Pi by 13. Now we know that cos Pi is equal to 0. Right? So is 0 cos Pi minus x gives us minus cos x. Also cos minus x is equal to cos x itself. Now if you carefully analyze this is cos minus x so we can write it as this cos minus x will be written as cos x only so it will be cos 8 Pi by 13 plus cos Pi minus x cos Pi minus x is minus cos x so here we'll write it as minus cos x and now here x is 10 Pi by 13 and similarly cos Pi minus now the x becomes 8 Pi by 13 so it will be written as minus cos x itself. So here we have minus cos 8 Pi by 13. Now on opening the brackets we have cos 10 Pi by 13 plus cos 8 Pi by 13 minus cos 10 Pi by 13 minus cos 8 Pi by 13 and now I think you must be realizing that you have reached to your answer as this will get cancelled out with this one and this will get cancelled out with this one and you left with the answer 0 that is equal to your RHS. So LHS is equal to RHS and now we can write down that hence we have proved the statement that was given to us, right? So I hope you enjoyed the session. Do remember the properties that you learned while starting off with the solution. Bye for now.