 Hello and welcome to the session. Let us understand the following problem tan inverse x is equal to half cos inverse of 1 minus x by 1 plus x where x belongs to closed interval 0 comma 1. Now the identity we will be using is 1 minus tan square theta by 1 plus tan square theta is equal to cos 2 theta by x is equal to tan square theta. This is our key idea. Now let us write the solution. Consider LHS. We have tan inverse root x put x is equal to tan square theta then theta is equal to tan inverse root x. Let us name this as 1. Now consider LHS half cos inverse of 1 minus x by 1 plus x which is equal to half cos inverse of 1 minus tan square theta by 1 plus tan square theta which is equal to half cos inverse of cos 2 theta using cos 2 theta is equal to 1 minus tan square theta by 1 plus tan square theta. Now which is equal to half into 2 theta which is equal to theta which is equal to tan inverse of root x by 1 which is equal to LHS therefore LHS is equal to RHS hence proved. You understood the problem. Bye and have a nice day.