 Hi and welcome to the session. Let us discuss the following question that says, proof the following, cos pi upon 4 minus x into cos pi upon 4 minus y minus sin pi upon 4 minus x into sin pi upon 4 minus y is equal to sin x plus y. So, let us start with the solution and here we will solve the LHS of the problem ensure that it is equal to the right hand side which is sin x plus y. So, the left hand side is cos pi upon 4 minus x into cos pi upon 4 minus y minus sin pi upon 4 minus x into sin pi upon 4 minus y. Now, let pi upon 4 minus x is equal to a and pi upon 4 minus y is equal to b. Then the LHS becomes cos a pi upon 4 minus x is a into cos b minus sin a into sin b and cos a cos b minus sin a sin b is cos a plus b and now let us substitute the values of a and b. So, we have cos pi upon 4 minus x plus b is pi upon 4 minus 5 which is further equal to cos adding pi by 4 and pi by 4 gives 2 pi by 4 minus taking common from minus x and minus y we have x plus y which is further equal to cos pi by 2 minus x plus y and since cos pi by 2 minus a is equal to sin a therefore, it can further be written as sin x plus y since a is x plus y and this is the right hand side of the given problem. Hence, we have left hand side is equal to right hand side and is proved. So, this completes the solution. Hope you enjoyed it. Take care and have a good day.