 Hello and welcome to the session. In this session, we are going to discuss the following question which says that first for x cross inverse of x plus cross inverse of 2x is equal to pi by 3 cross of a minus b is given by cross of a into cross of b plus sin of a into sin of b and cross inverse of x is equal to sin inverse of square root of 1 minus x square with this key idea let us proceed with the solution we are given that cross inverse of x plus cross inverse of 2x is equal to pi by 3 which can be written as cross inverse of 2x is equal to pi by 3 minus cross inverse of x which implies that 2x is equal to cross of pi by 3 minus cross inverse of x and we know that cross of a minus b is equal to cross of a into cross of b plus sin of a into sin of b therefore 2x is equal to cross of pi by 3 into cross of cross inverse of x plus sin of pi by 3 into sin of cross inverse of x which implies that 2x is equal to cross pi by 3 is 1 by 2 into cross of cross inverse of x is x plus sin pi by 3 which is square root of 3 by 2 into sin of cross inverse of x which is equal to sin of sin inverse of square root of 1 minus x square since cross inverse of x is equal to sin inverse of square root of 1 minus x square which can be written as 2x is equal to x by 2 plus square root of 3 by 2 into square root of 1 minus x square since sin of sin inverse of x is equal to x for all x belonging to the closed interval minus 1 to 1 then we get 2x is equal to x plus square root of 3 into 1 minus x square whole upon 2 that is 4x is equal to x plus square root of 3 into 1 minus x square which implies that 4x minus x that is 3x is equal to square root of 3 into 1 minus x square now on squaring both sides we get 3x square is equal to 3 into 1 minus x square which implies that 9x square is equal to 3 minus 3x square that is 9x square plus 3x square is equal to 3 which is equal to 12x square is equal to 3 that is x square is equal to 3 by 12 that is 1 by 4 we can write x is equal to plus minus 1 by 2 if we put x equal to 1 by 2 in the given equation that is cos inverse of x plus cos inverse of 2x is equal to pi by 3 taking the left hand side we get cos inverse of 1 by 2 plus cos inverse of 2 into 1 by 2 which is equal to cos inverse of 1 by 2 plus cos inverse of 1 that is pi by 3 plus 0 which is equal to pi by 3 and equal to the right hand side of the equation it shows that x equal to 1 by 2 that is 5 the equation now if we put x equal to minus 1 by 2 in the given equation that is cos inverse of x plus cos inverse of 2x is equal to pi by 3 on taking the left hand side we get cos inverse of minus 1 by 2 plus cos inverse of 2 into minus 1 by 2 which can be written as cos inverse of minus 1 by 2 plus cos inverse of minus 1 and we know that cos inverse of minus x is given by pi minus cos inverse of x for all x belonging to the close interval minus 1 to 1 therefore we have cos inverse of minus 1 by 2 is equal to pi minus cos inverse of 1 by 2 plus cos inverse of minus 1 can be written as pi minus cos inverse of 1 which is equal to 2 pi minus pi by 3 minus 0 which is equal to 5 pi by 3 that is not equal to the right hand side of the equation therefore x equal to minus 1 by 2 does not satisfy the equation x equal to 1 by 2 is the required answer please find session hope you enjoyed this session