 Hello and welcome to the session. Let us understand the following problem today. Write the following function in the simplest form. We have tan inverse under root 1 plus x square minus 1 by x where x is not equal to 0. Now let us write the solution. Let x is equal to tan theta then under root 1 plus x square is equal to under root 1 plus tan square theta which is equal to sec theta. Let us name this as 1. Now tan inverse of under root 1 plus x square minus 1 whole divided by x will be equal to tan inverse of sec theta minus 1 by tan theta. This is by using 1 which is equal to tan inverse of 1 by cos theta minus 1 whole divided by sin theta by cos theta which is equal to tan inverse of 1 minus cos theta by sin theta which is equal to tan inverse of 2 sin square theta by 2 by 2 sin theta by 2 cos theta by 2. Now this 2 gets cancelled and this sin theta by 2 gets cancelled by 1 sin theta by 2. So we are left with tan inverse of sin theta by 2 divided by cos theta by 2 which is equal to tan inverse of tan theta by 2 which is equal to theta by 2 which is equal to half tan inverse of x because x is equal to tan theta which implies x theta is equal to tan inverse of x. Therefore tan inverse of under root 1 plus x square minus 1 whole divided by x where x is not equal to 0 is equal to half tan inverse of x hence proved.