 Hello and welcome to the session. In this session we are going to discuss the following question which says that算 for x tan inverse of 2x minus 1 upon 2x minus 3 plus tan inverse of 2x plus 1 upon 2x plus 3 is equal to pi by 4 tan inverse of x minus tan inverse of y is equal to tan inverse of x minus y upon 1 plus x y divided x into y is greater than minus 1. With this key idea we shall proceed with the solution. We are given that tan inverse of 2x minus 1 upon 2x minus 3 plus tan inverse of 2x plus 1 upon 2x plus 3 is equal to pi by 4 which can also be written as tan inverse of 2x minus 1 upon 2x minus 3 plus tan inverse of 2x plus 1 upon 2x plus 3 is equal to tan inverse of 1 since tan pi by 4 is equal to 1 which implies that tan inverse of 1 is equal to pi by 4 or we can write tan inverse of 2x minus 1 upon 2x minus 3 is equal to tan inverse of 1 minus tan inverse of 2x plus 1 upon 2x plus 3 tan inverse of x minus tan inverse of y is given by tan inverse of x minus y upon 1 plus x y divided x into y is greater than minus 1. Therefore we get tan inverse of 1 minus 2x plus 1 upon 2x plus 3 pi upon 1 plus 1 into 2x plus 1 upon 2x plus 3 which is equal to tan inverse of 2x plus 3 minus 2x minus 1 upon 2x plus 3 whole upon 1 plus 2x plus 1 upon 2x plus 3 which is equal to tan inverse of 2 upon 2x plus 3 whole upon 2x plus 3 plus 2x plus 1 upon 2x plus 3 which is equal to tan inverse of 2 upon 2x plus 3 whole upon 4x plus 4 upon 2x plus 3 which is equal to tan inverse of 2 upon 4x plus 4 which is further equal to tan inverse of 1 upon 2x plus 2. Therefore we have tan inverse of 2x minus 1 upon 2x minus 3 is equal to tan inverse of 1 upon 2x plus 2. Now the application of the formula tan inverse of x minus tan inverse of y equal to tan inverse of x minus y upon 1 plus x y is justified 1 into 2x plus 1 upon 2x plus 3 is greater than minus 1 plus 1 is greater than minus 2x minus 3 or we can say 4x is greater than minus 4 which implies that x is greater than minus 1. Now we have tan inverse of 2x minus 1 upon 2x minus 3 is equal to tan inverse of 1 upon 2x plus 2 which implies that 2x minus 1 upon 2x minus 3 is equal to 1 upon 2x plus 2 which can be written as 2x minus 1 into 2x plus 2 is equal to 2x minus 3 that is 4x square plus 4x minus 2x minus 2 is equal to 2x minus 3 which implies that 4x square plus 2x minus 2 is equal to 2x minus 3 is greater than 4x square is equal to 2x minus 3 minus 2x plus 2 therefore 4x square is equal to minus 1 which implies that x square is equal to minus 1 by 4 that is x is given by minus plus 1 by 2 since minus 1 by 2 and 1 by 2 are greater than minus 1 therefore both are admissible hence the value of x is given by minus plus 1 by 2 which is the required answer this completes our session but we will drive this session