 Hello and welcome to the session. Let us understand the following problem today. Prove that tan inverse 1 by 5 plus tan inverse 1 by 7 plus tan inverse 1 by 3 plus tan inverse 1 by 8 is equal to pi by 4. The formula that we will be using in this problem will be tan inverse a plus tan inverse b is equal to tan inverse a plus b by a 1 minus a b. Where a b is less than 1, let us name it as a. This is our key idea. Now let us write the solution. Consider tan inverse 1 by 5 plus tan inverse 1 by 7. Now using the identity a, we can write tan inverse 1 by 5 plus tan inverse 1 by 7 is equal to tan inverse 1 by 5 plus 1 by 7 by 1 minus 1 by 5 into 1 by 7 which is equal to tan inverse 12 by 35 by 34 by 35 which is equal to tan inverse 12 by 34 which is equal to tan inverse 6 by 17. Let us name it as 1. Now consider tan inverse 1 by 3 plus tan inverse 1 by 8. Now using identity a, we can write tan inverse 1 by 3 plus tan inverse 1 by 8 is equal to tan inverse 1 by 3 plus 1 by 8 divided by 1 minus 1 by 3 into 1 by 8 which is equal to tan inverse 11 by 24 whole divided by 23 by 24 which is equal to tan inverse of 11 by 23. Let us name it as 2. Now thus from 1 and 2 we can write tan inverse 1 by 5 plus tan inverse 1 by 7 plus tan inverse 1 by 3 plus tan inverse 1 by 8 is equal to tan inverse 6 by 17 plus tan inverse 11 by 23. Now again using the identity a, we can write this as tan inverse 6 by 17 plus tan inverse 11 by 23 can be written as tan inverse 6 by 17 plus 11 by 23 by 1 minus 6 by 17 into 11 by 23 which is equal to tan inverse on solving we get 325 by 325 which is equal to tan inverse of 1 which is equal to pi by 4 thus LHS is equal to RHS hence proved. I hope you understood this problem. Bye and have a nice day.