 Hello and welcome to the session. In this session, we are going to discuss the following question which says that if cross inverse of 1 by square root of 1 plus a square plus cross inverse of 1 by square root of 1 plus b square plus cross inverse of 1 by square root of 1 plus c square is equal to 2 pi then show that a plus b the whole square is equal to a b c minus c the whole square cross of a plus b is equal to cross of a into cross of b minus sin of a into sin of b with this key idea let us proceed with the solution. We are given that cross inverse of 1 upon square root of 1 plus a square plus cross inverse of 1 upon square root of 1 plus b square plus cross inverse of 1 upon square root of 1 plus c square is equal to 2 pi let us Let us assume cos inverse of 1 upon square root of 1 plus a square dA cos inverse of 1 upon square root of 1 plus d square dB and cos inverse of 1 upon square root of 1 plus c square dC cos inverse of 1 by square root of 1 plus a square equal to a implies that that cos of a is equal to 1 upon square root of 1 plus a square. Cos inverse of 1 by square root of 1 plus b square equal to b implies that cos of b is equal to 1 upon square root of 1 plus b square and cos inverse of 1 upon square root of 1 square equal to c implies that cos of c is equal to 1 upon square root of 1 plus c square. Now we have cos of a is equal to 1 upon square root of 1 plus a square in a triangle p q r if a is the angle between the lines p r and r q then cos of angle a is given by base upon hypotenuse that is r q upon p r which is equal to 1 upon square root of 1 plus a square if we are given the value of the base r q as 1 and the value of the hypotenuse r p as square root of 1 plus a square then we can find the value of the perpendicular p q by using the Pythagoras theorem. By Pythagoras theorem we have perpendicular p q is given by square root of hypotenuse p r square minus base r q square which is equal to square root of square root of 1 plus a square then a square will be a square minus 1 square which is equal to square root of 1 plus a square minus 1 that is square root of a square which is equal to a so perpendicular p q is given by a and we know that sin of a is given by perpendicular upon hypotenuse that is p q upon r p which is equal to a upon square root of 1 plus a square also we have cos of b is equal to 1 upon square root of 1 plus b square in a triangle p q r if b is the angle between the lines p r and r q then cos of b is given by base upon hypotenuse that is r q upon r p which is equal to 1 upon square root of 1 plus b square if we are given base r q as 1 and hypotenuse r p as square root of 1 plus b square then we can find the value of the perpendicular p q by using Pythagoras theorem. By Pythagoras theorem we have perpendicular p q is equal to square root of hypotenuse r p square minus base r q square which is equal to square root of square root of 1 plus b square g whole square minus 1 square that is square root of 1 plus b square minus 1 which is equal to square root of b square that is b therefore the value of perpendicular p q is b and we know that sin of b is given by perpendicular upon hypotenuse that is p q upon p r which is equal to b upon square root of 1 plus b square we have assumed cos of 1 upon square root of 1 plus a square at a cos of 1 upon square root of 1 plus b square as b and cos of 1 upon square root of 1 plus c square as c putting these values in the given expression we get a plus b plus c is equal to 2 pi all we can write a plus b is equal to 2 power minus c taking cos on both the sides we get cos of a plus b is equal to cos of 2 pi minus c cos of a plus b is equal to cos of a into cos of b minus sin of a into sin of b which implies that cos of a into cos of b minus sin of a into sin of b is equal to cos of c as cos of 2 pi minus c is equal to cos of c now we have the values of cos a cos b cos c sin a and sin b putting all these values in the equation we get 1 upon square root of 1 plus a square into 1 upon square root of 1plus b square minus a upon square root of 1plus a square into b upon b upon square root of 1 plus b square is equal to 1 upon square root of 1 plus c square which implies that 1 upon square root of 1 plus a square into 1 plus b square minus a b upon square root of 1 plus a square into 1 plus b square is equal to 1 upon square root of 1 plus c square which can be written as 1 minus a b upon square root of 1 plus a square into 1 plus b square is equal to 1 upon square root of 1 plus c square or we can also write 1 minus a b is equal to square root of 1 plus a square into 1 plus b square upon 1 plus c square on square in both sides we get 1 minus a b d whole square is equal to 1 plus a square into 1 plus b square upon 1 plus c square which implies that 1 plus a square b square minus 2 a b is equal to 1 plus a square into 1 plus b square whole upon 1 plus c square which implies that 1 plus a square b square minus 2 a b into 1 plus c square is equal to 1 plus a square plus b square plus a square b square on solving server we get 1 plus a square b square minus 2 a b plus c square plus a square b square c square minus 2 a b c square is equal to 1 plus a square plus b square plus a square b square now taking all the terms on the left hand side we get 1 plus a square b square minus 2 a b plus c square plus a square b square c square minus 2 a b c square minus 1 minus a square minus b square minus a square b square is equal to 0 which implies that a square b square c square minus 2 a b c square minus 2 a b plus c square minus a square minus b square is equal to 0 which can also be written as a square b square c square minus 2 a b c square plus c square minus a square plus b square plus 2 a b is equal to 0 which can also be written as a square b square c square minus a b c square minus a b c square plus c square minus a square plus b square plus 2 a b equal to 0 now taking a b c common from these 2 terms we get a b c into a b c minus c taking minus c common from these 2 terms we get a b c minus c minus a square plus b square plus 2 a b is equal to 0 which can also be written as a b c minus c is equal to a square plus b square plus 2 a b that is a b c minus c b whole square is equal to a plus b b whole square as a plus b whole square is equal to a square plus b square plus 2 a b hence we have proved that a plus b whole square is equal to a b c minus c b whole square this completes our session hope you enjoyed this session