 Hello and welcome to the session in this session we are going to discuss the following question which says that show that cos inverse of 4 by square root of 17 plus cos inverse of 9 by square root of 85 is equal to 5 minus cos inverse of 2 by square root of 5. We know that cos inverse of x plus cos inverse of y is equal to 5 minus cos inverse of xy minus square root of 1 minus x square into square root of 1 minus y square. If both x and y are greater than 0 and x square plus y square is greater than 1, with this key idea let us proceed with the solution. We need to prove that cos inverse of 4 upon square root of 17 plus cos inverse of 9 upon square root of 85 is equal to 5 minus cos inverse of 2 upon square root of 5. On the left hand side we have cos inverse of 4 by square root of 17 plus cos inverse of 9 by square root of 85. Now we know that cos inverse of x plus cos inverse of y is given by pi minus cos inverse of xy minus square root of 1 minus x square into square root of 1 minus y square provided both x and y are greater than 0 and x square plus y square is greater than 1. Here we assume 4 by square root of 17 as x and 9 by square root of 85 as y and we know that both 4 upon square root of 17 and 9 upon square root of 85 are greater than 0. Also the value of the expression 4 by square root of 17 the whole square plus 9 by square root of 85 the whole square is given by 16 by 17 plus 81 by 85 that is 80 plus 81 by 85 which is equal to 161 by 85 is greater than 1. Therefore we can write cos inverse of 4 by square root of 17 plus cos inverse of 9 by square root of 85 is equal to pi minus cos inverse of 4 by square root of 17 into 9 by square root of 85 minus square root of 1 minus 4 by square root of 17 square into square root of 1 minus 9 upon square root of 85 square which is equal to pi minus cos inverse of 36 upon square root of 17 into 85 minus square root of 1 minus 16 by 17 into square root of 1 minus 81 by 85 which is further equal to pi minus cos inverse of 36 upon square root of 1445 minus square root of 17 minus 16 by 17 into square root of 85 minus 81 by 85 which gives cos inverse of 36 upon 17 into square root of 5 minus square root of 1 by 17 into square root of 4 by 85 which is equal to pi minus cos inverse of 36 upon 17 square root of 5 minus square root of 4 upon 17 into 85 which is equal to pi minus cos inverse of 36 upon 17 into square root of 5 minus 2 upon 17 into square root of 5 which is equal to pi minus cos inverse of 36 minus 2 upon 17 into square root of 5 that is pi minus cos inverse of 34 by 17 into square root of 5 which is equal to pi minus cos inverse of 2 by square root of 5 which is equal to the right-hand side of the equation. Hence we have proved that cos inverse of 4 by square root of 17 plus cos inverse of 9 by square root of 85 is equal to pi minus cos inverse of 2 by square root of 5 That completes our session. Hope you enjoyed this session.