 Hello and welcome to the session. In this session we are going to discuss the following question which says that evaluate 4 tan inverse of 1 by 7 minus tan inverse of 3 by 4 plus tan inverse of 16 by 63. We know that 2 tan inverse of x is equal to tan inverse of 2x upon 1 minus x square if x square is less than 1 and tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus x y provided x into y should be greater than minus 1. With this key idea let us proceed with the solution. We are given the expression 4 tan inverse of 1 by 7 minus tan inverse of 3 by 4 plus tan inverse of 16 by 63 which can be written as 2 into 2 tan inverse of 1 by 7 minus of tan inverse of 3 by 4 minus 16 by 63 First let us solve 2 into 2 tan inverse of 1 by 7. From the key idea we know that 2 tan inverse of x is given by tan inverse of 2x upon 1 minus x square if x square is less than 1. Here we assume x as 1 by 7 and x square should be less than 1 that is 1 by 7 square which is equal to 1 by 49 should be less than 1 which is true. Therefore we can write 2 tan inverse of 1 by 7 as tan inverse of 2 by 7 upon 1 minus 1 by 7 square which is equal to 2 by 7 upon 1 minus 1 by 49 that is tan inverse of 2 by 7 upon 49 minus 1 upon 49 which is equal to tan inverse of 2 by 7 upon 48 by 49 that is tan inverse of 7 into 49 by 48 is equal to tan inverse of 2 by 24. Therefore 2 tan inverse of 1 by 7 will solve 7 by 24 and the expression 2 into 2 tan inverse of 1 by 7 can be written as 2 tan inverse of 7 by 24 and using the same formula that is 2 tan inverse of x is equal to tan inverse of 2x upon 1 minus x square if x square is less than 1. Here we can assume x as 7 by 24 and x square should be less than 1 that is 7 by 24 square should be less than 1 which is equal to 79 upon 576 which is less than 1. Therefore 2 tan inverse of 7 by 24 can be written as tan inverse of 2 into 7 by 24 upon 1 minus 7 by 24 square which is equal to tan inverse of by 12 upon 49 upon 576 which is equal to tan inverse of 7 by 12 upon 576 minus 49 upon 576 which is equal to tan inverse of 7 by 12 upon 527 upon 576 can be written as tan inverse of 7 by 12 into 576 upon 527 is equal to tan inverse of 336 upon 527 into 2 tan inverse of 1 by 7 is equal to tan inverse of 336 upon 527 all we can write 4 tan inverse of 1 by 7 is equal to tan inverse of 336 upon 527 which has solved the expression tan inverse of 3 by 4 minus tan inverse of 16 by 63 that is we have tan inverse of 3 by 4 minus tan inverse of 16 by 63 and we know that tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus x y provided x into y is greater than minus 1 here we can assume x as 3 by 4 and y as 16 by 63 and we know that x into y should be greater than minus 1 that is 3 by 4 into 16 by 63 that is 48 upon 252 should be greater than minus 1 which is true so we can write tan inverse of 3 by 4 minus tan inverse of 16 by 63 is equal to tan inverse of 3 by 4 minus 16 by 63 whole upon 1 plus 3 by 4 into 16 by 63 which is equal to tan inverse of 189 minus 64 upon 252 whole upon 1 plus 48 upon 252 which is further equal to tan inverse of 125 upon 252 whole upon 252 plus 48 upon 252 which is equal to tan inverse of 125 upon 300 into 252 upon 252 which is equal to tan inverse of by 12 therefore the value of the expression tan inverse of 3 by 4 minus tan inverse of 16 by 63 is equal to tan inverse of 5 by 12 now we have 2 into 2 tan inverse of 1 by 7 as tan inverse of 336 upon 527 tan inverse of 3 by 4 minus tan inverse of 16 by 63 is equal to tan inverse of 5 by 12 substituting these values in the given expression we get tan inverse of 336 upon 527 minus tan inverse of 5 by 12 and using the same formula again that is tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus xy provided x into y is greater than minus 1 here we can assume x of 336 upon 527 and y of 5 by 12 and we know that x into y should be greater than minus 1 that is 336 upon 527 into 5 by 12 should be greater than minus 1 which is true therefore we can write tan inverse of 336 upon 527 minus tan inverse of 5 by 12 as tan inverse of 336 upon 527 minus 5 by 12 upon 1 plus 336 upon 527 into 5 by 12 which is equal to tan inverse of 4032 minus 2635 upon 6324 divided by 6324 plus 1680 upon 6324 which is equal to tan inverse of 1397 upon 8004 therefore the value of the expression for tan inverse of 1 by 7 minus tan inverse of 3 by 4 plus tan inverse of 16 by 63 is given by tan inverse of 1397 upon 8004 which is the required answer this completes our session hope you enjoyed this session