 Hello and welcome to the session. In this session we are going to discuss the following question which says that evaluate tan of 2 tan inverse of 3 by 4 minus pi by 4. We know that 2 tan inverse of x is equal to tan inverse of 2x upon 1 minus x square where modulus of x is less than 1. Also tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus xy xy is greater than minus 1. With this key idea we shall proceed with the solution. We have the expression tan of 2 tan inverse of 3 by 4 minus pi by 4. First let us solve 2 tan inverse of 3 by 4. From the key idea we know that 2 tan inverse of x is equal to tan inverse of 2x upon 1 minus x square where modulus of x is less than 1. Therefore 2 tan inverse of 3 by 4 will be equal to tan inverse of 2 into 3 by 4 upon 1 minus 3 by 4 the whole square. Also modulus of 3 by 4 is less than 1. So we have 2 tan inverse of 3 by 4 is equal to tan inverse of 3 by 2 upon 1 minus 9 upon 16 which implies that 2 tan inverse of 3 by 4 is equal to tan inverse of 3 by 2 by 16 minus 9 by 16 which further implies that 2 tan inverse of 3 by 4 is equal to tan inverse of 3 by 2 upon 7 by 16 which is equal to 2 tan inverse of 3 by 4 is equal to tan inverse of 3 by 2 into 16 by 7 which implies that 2 tan inverse of 3 by 4 is equal to tan inverse of 24 by 7. Now replace the value of 2 tan inverse of 3 by 4 with tan inverse of 24 by 7. In the given expression we get tan of tan inverse of 24 by 7 minus pi by 4 which we can write as tan of tan inverse of 24 by 7 minus tan inverse of 1 as tan pi by 4 is equal to 1 so tan inverse of 1 is equal to pi by 4. Mark this equation as A. Now let us simplify tan inverse of 24 by 7 minus tan inverse of 1 using the key idea which says that tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus xy if xy is greater than minus 1. So we have tan inverse of 24 by 7 minus tan inverse of 1 is equal to tan inverse of 24 by 7 minus 1 upon 1 plus 24 by 7 into 1 and 24 by 7 is greater than minus 1 which implies that tan inverse of 24 by 7 minus tan inverse of 1 is equal to tan inverse of 24 minus 7 by 7 upon 1 plus 24 by 7 which further implies that tan inverse of 24 by 7 minus tan inverse of 1 is equal to tan inverse of 17 upon 7 whole upon 7 plus 24 by 7 which implies that tan inverse of 24 by 7 minus tan inverse of 1 is equal to tan inverse of 17 upon 31. Now substitute the value of tan inverse of 24 by 7 minus tan inverse of 1 in A we get an of tan inverse of 17 by 31 which is equal to 17 upon 31. Therefore the value of the expression tan of 2 tan inverse of 3 by 4 minus pi by 4 is equal to 17 by 31 which is the required answer. This completes our session hope you enjoyed this session.