 Hello and welcome to the session. In this session we are going to discuss the following question which says that if sign inverse of x by a minus sign inverse of y by b is equal to theta, x by a and y by b are greater than equal to 0 and v square x square plus a square y square is less than equal to a square b square, minus that a square b square sine square theta plus 2ab y square root of a square minus x square sine theta plus y square a square minus x square b square is equal to 0 and a square b square sine square theta minus 2ab x square root of b square minus y square sine theta plus x square b square minus y square a square is equal to 0. We know that sign inverse of x minus sign inverse of y is equal to sign inverse of x into square root of 1 minus y square minus y into square root of 1 minus x square if both x and y are greater than equal to 0 and x square plus y square is less than equal to 1. With this key idea we shall proceed with the solution. We have given the equation sign inverse of x by a minus sign inverse of y by b is equal to theta. Also we are given that x by a and y by b are greater than equal to 0 and b square x square plus a square y square is less than equal to a square b square. So we have b square x square plus a square y square is less than equal to a square b square. Now dividing both the sides of the equation by a square b square we have b square x square upon a square b square plus a square y square upon a square b square is less than equal to a square b square upon a square b square which is equal to x square by a square plus y square by b square is less than equal to 1 which can also be written as x by a the whole square plus y by b the whole square is less than equal to 1. From the key idea we know that sign inverse of x minus sign inverse of y is given by sign inverse of x into square root of 1 minus y square minus y into square root of 1 minus x square provided both x and y are greater than 0 and x square plus y square is less than equal to 1. And here we have x by a and y by b greater than equal to 0 also x by a the whole square plus y by b the whole square is less than equal to 1. So the expression sign inverse of x by a minus sign inverse of y by b can be written as sign inverse of x by a into square root of 1 minus y by b square minus y by b into square root of 1 minus x by a square which is equal to theta. This implies that sign inverse of x by a into square root of 1 minus y square upon b square minus y by b into square root of 1 minus x square by a square is equal to theta that is sign inverse of x by a into square root of b square minus y square upon b square minus y by b into square root of a square minus x square by a square is equal to theta which further implies that sign inverse of x by a b into square root of b square minus y square minus y upon a b into square root of a square minus x square is equal to theta. This implies that x upon a b into square root of b square minus y square minus y upon a b into square root of a square minus x square is equal to sign of theta. Now this equation as a it can also be written as x upon a b into square root of b square minus y square is equal to sign theta plus y upon a b into square root of a square minus x square. Now, square in both the sides we get x upon a b into square root of b square minus y square the whole square equal to sign of theta plus y upon a b into square root of a square minus x square the whole square which implies that x square by a square b square into b square minus y square is equal to sign square theta plus y square upon a square b square into a square minus x square plus 2 y upon a b into square root of a square minus x square sign theta on taking the LCM on the right hand side of the equation we get x square by a square b square into b square minus y square is equal to a square b square into sign square theta plus y square into a square minus x square plus 2 a b y into a square minus x square in the square root into sign theta whole upon a square b square. Now, multiplying both the sides by a square b square we get x square into b square minus y square is equal to a square b square sign square theta plus y square into a square minus x square plus 2 a b y into a square minus x square sign of theta which further implies that x square b square minus x square y square is equal to a square b square sign square theta plus y square a square minus y square x square plus 2 a b y into square root of a square minus x square into sign theta Now, taking all the terms on the right hand side we get a square b square sign square theta plus a square y square minus x square y square plus 2 a b y into square root of a square minus x square into sign theta minus of x square b square plus x square y square is equal to 0 which implies that a square b square sign square theta plus a square y square plus 2 a b y into square root of a square minus x square sign theta minus x square b square is equal to 0 Hence, prove the result that is a square b square sign square theta plus 2 a b y square root of a square minus x square sign theta plus y square a square minus x square b square is equal to 0 Now, consider the equation a which can be written as x into square root of b square minus y square upon a b minus of sign theta is equal to y into square root of a square minus x square by a b Now, squaring both sides we get simple square root of b square minus y square by a b minus sign theta b whole square is equal to y into square root of a square minus x square by a b b whole square which implies that x square into b square minus y square upon a square b square plus sign square theta minus 2 into x into square root of b square minus y square upon a b into sign theta is equal to y square into a square minus x square by a square b square Now, taking the LCM on the left hand side we get x square into b square minus y square plus a square b square sign square theta minus 2 a b x into b square minus y square in the square root into sign theta upon a square b square is equal to y square into a square minus x square whole upon a square b square Now, multiplying both the sides by a square b square we get x square into b square minus y square plus a square b square sign square theta minus 2 a b x into square root of b square minus y square sign theta is equal to y square into a square minus x square which implies that x square b square minus x square y square plus a square b square sign square theta minus 2 a b x into square root of b square minus y square sign theta is equal to y square a square minus y square x square Now, taking all the terms on the left hand side of the equation we get x square b square minus x square y square plus a square b square sign square theta minus 2 a b x into square root of b square minus y square sign theta minus of y square a square plus y square x square is equal to 0 on further solving we get x square b square plus a square b square sign square theta minus 2 a b x into square root of b square minus y square sign theta minus y square a square is equal to 0 Hence, prove the result that the a square b square sign square theta minus 2 x a b into square root of b square minus y square sign theta plus x square b square minus y square a square is equal to 0 This completes our session. Hope you enjoyed this session.