 Hello and welcome to the session. In this session we are going to discuss the following question which says that evaluate cos square of tan inverse of x plus x square of cot inverse of y. Let's start the solution. We are given the expression cos square of tan inverse of x plus x square of cot inverse of y which can be written as cos of tan inverse of x the whole square plus sec of cot inverse of y the whole square. First let us solve cos of tan inverse of x let's tan inverse of x be theta which implies that x is equal to tan of theta and we can write tan theta is equal to x in a triangle ADP if theta is the angle between the lines AC and CD then tan of angle theta is given by perpendicular upon base that is AD upon BC which is equal to x by 1. Now if perpendicular AB is given by x and base BC is given by 1 then we can find the value of the hypotenuse AC by using Pythagoras theorem. By Pythagoras theorem we have hypotenuse AC is given by square root of perpendicular AB square plus base BC square which is equal to square root of x square plus 1 square which is given by square root of x square plus 1. So hypotenuse AC is equal to square root of x square plus 1. We know that cos of theta is given by base upon hypotenuse that is BC upon AC which is equal to 1 upon square root of x square plus 1 which implies that theta is given by cos inverse of 1 upon square root of x square plus 1. Now we have tan inverse of x is equal to theta and theta is equal to cos inverse of 1 upon square root of x square plus 1 which implies that tan inverse of x is equal to cos inverse of 1 upon square root of x square plus 1. Therefore the expression cos of can be written as cos of cos inverse of 1 upon square root of x square plus 1 which can be written as 1 upon square root of x square plus 1. Therefore cos of tan inverse of x is equal to 1 upon square root of x square plus 1. Now let us self set cos inverse of y let cos inverse of y be phi which implies that y is equal to cos of phi as we can write cos of phi is equal to y. In a triangle PQR if phi is the angle between the lines PR and RQ then quarter phi is given by base upon perpendicular that is RQ upon PQ which is equal to y by 1. If we are given perpendicular PQS1 and base RQSy then we can find the value of the hypotenuse PR by using Pythagoras theorem. By Pythagoras theorem we have hypotenuse RP is given by square root of perpendicular PQ square plus base RQ square which is equal to square root of 1 square plus y square which is further equal to square root of 1 plus y square. Therefore the value of the hypotenuse RP is equal to square root of 1 plus y square. Now we know that second side is given by hypotenuse type base that is RP by RQ which is equal to square root of 1 plus y square by y which implies that phi is given by second whole source square root of 1 plus y square upon y. Here we have quarter inverse of y is equal to phi and phi is equal to second whole source square root of 1 plus y square by y which implies that quarter inverse of y is equal to second whole source square root of 1 plus y square by y. Therefore the expression second whole source square root of 1 plus y square by y which can be written as square root of 1 plus y square by y. Therefore third whole source quarter inverse of y is equal to square root of 1 plus y square upon y. Now we have quarter source tan inverse of x is equal to 1 upon square root of x square plus 1 and second whole source quarter inverse of y is equal to square root of 1 plus y square whole upon y. Now substituting these two values in the given expression we get quarter source tan inverse of x the whole square plus third whole source quarter inverse of y the whole square is equal to 1 upon square root of x square plus 1 the whole square plus square root of 1 plus y square by y the whole square which is equal to 1 upon x square plus 1 plus 1 plus y square upon y square. On taking the LCM we get y square plus 1 plus y square into x square plus 1 whole upon y square into x square plus 1. So we get quarter tan inverse of x the whole square plus third whole source quarter inverse of y the whole square is equal to y square plus 1 plus y square into x square plus 1 whole upon y square into x square plus 1. Now we can also write quarter square root tan inverse of x plus 6 square root quarter inverse of y is equal to y square plus 1 plus y square into x square plus 1 whole upon y square into x square plus 1 which will be required answer this completes our session hope you enjoyed this session.