 Hi and welcome to the session. Let us prove the following that cos 3 pi upon 2 plus x into cos 2 pi plus x square bracket cot 3 pi upon 2 minus x plus cot 2 pi plus x is equal to 1. So, let us begin with the solution and we will solve the left hand side of this problem and show that its value is equal to 1. So, left hand side is cot 3 pi upon 2 plus x into cos 2 pi plus x square bracket cot 3 pi upon 2 minus x plus cot 2 pi plus x. Now first let us learn some simple identities. First is cos 3 pi upon 2 plus x is equal to sin x. Since this angle lies in the fourth quadrant and in the fourth quadrant cos is a positive function. So, we have sin x. Since if we have cos n pi upon 2 plus x where n is a hot number then cos changes to sin. Second is cos x plus x. Now this lies in the first quadrant. So, this is cos x. Since in the first quadrant all the functions are positive. Then we have cot 3 pi upon 2 minus x which is equal to tan x and the last one is cot 2 pi plus x which is equal to cos x. So, with the help of these functions l h is of the above problem given as plus cos 3 pi upon 2 plus x is equal to sin x. Then we have cos 2 pi plus x which is cos x and the square bracket we have cot 3 pi upon 2 minus x whose value is tan x plus cot 2 plus x is cot which is further equal to sin x into cos x and in the bracket tan x will be equal to sin x upon cos x and cot x will be equal to cos x upon sin x. Now solving the square bracket is sin x into cos x and now I will show you sin x and cos x is cos x sin x and the numerator we have sin square x plus cos square x which is further equal to sin x into cos x divided by cos x into sin x and the bracket we have sin square x plus cos square x. Now since sin square x plus cos square x is equal to 1 therefore we can further written as 2 cancels out we have 1 and sin square x plus cos square x is 1 so 1 into 1 is 1 which is the right hand side of the above problem plus we have cos 3 pi upon 2 plus x into cos 2 pi plus x square bracket cot 3 pi upon 2 minus x plus cot 2 pi plus x is equal to 1 hence proved. So this completes the solution hope you enjoyed it take care.