 Hello and welcome to the session. In this session, we discussed the following question which says if cossack phi minus sine phi is equal to n cube and sec phi minus coss phi is equal to n cube, prove that n square into n square into n square plus n square is equal to 1. So, let's see its solution now. We are given that cossack phi minus sine phi is equal to m cube and sec phi minus coss phi is equal to n cube and we need to prove that m square into n square into n square plus n square is equal to 1. Now, first consider cossack phi minus sine phi equal to m cube. So, we have m cube is equal to 1 upon sine phi minus sine phi. Since we know that sine theta is equal to 1 upon cossack theta. So, cossack phi would be equal to 1 upon sine phi. So, this gives us m cube is equal to sine phi in the denominator and in the numerator 1 minus sine square phi in the numerator. So, further we get m cube is equal to coss square phi upon sine phi since we know that sine square theta plus coss square theta is equal to 1. So, 1 minus sine square theta is equal to coss square theta. So, we have written 1 minus sine square phi is coss square phi. So, we have got n cube equal to coss square phi upon sine phi. So, from here we can say that m is equal to coss 2 upon 3 phi upon sine 1 upon 3 phi. So, we have got the values of n cube and m. Now, next we have n cube is equal to sec phi minus cos phi. This means n cube is equal to 1 upon cos phi minus cos phi. Since we know that cos theta is equal to 1 upon sec theta. So, we can write sec phi as 1 upon cos phi. Further we get n cube is equal to cos phi in the denominator. In the numerator we have 1 minus cos square phi. Thus, we get n cube is equal to sine square phi upon cos phi since we know that sine square theta plus cos square theta is equal to 1. So, sine square theta is equal to 1 minus coss square theta. And hence we have written 1 minus cos square phi as sine square phi. From here we get n is equal to sine 2 upon 3 phi upon cos 1 upon 3 phi. So, now we have got the values for n cube and n. We were supposed to prove that m square into n square plus n square is equal to 1. So, now we shall consider the LHS that is equal to m square into n square into m square plus n square which is equal to m to the power 4 into n square plus n square into n to the power 4. Or you can say this is equal to m cube into m into n square plus n cube into n into n square. So, now we can substitute the values for m cube n cube m n m square and n square. So, this is equal to now the value of m cube is cos square phi upon sine phi. So, this would be equal to cos square phi upon sine phi this into the value of m which is cos 2 upon 3 phi upon sine 1 upon 3 phi and this into n square. Now, the value of n is equal to sine 2 upon 3 phi upon cos 1 upon 3 phi. So, n square is given by sine 4 upon 3 phi upon cos 2 upon 3 phi plus n cube. Now, the value of n cube is sine square phi upon cos phi this into n which is sine 2 upon 3 phi upon cos 1 upon 3 phi this into n square. Now, that we have n is equal to cos 2 upon 3 phi upon sine 1 upon 3 phi. So, n square is given by cos 4 upon 3 phi upon sine 2 upon 3 phi. Now, this is further equal to cos square phi upon sine phi this into the sine 4 upon 3 phi upon sine 1 upon 3 phi is given as sine phi and this cos 2 upon 3 phi and cos 2 upon 3 phi cancels. Now, this plus sine square phi upon cos phi this into the cos 4 upon 3 phi upon cos 1 upon 3 phi is given as cos phi and sine 2 upon 3 phi sine 2 upon 3 phi cancels. Now, here sine phi and sine phi cancels cos phi and cos phi cancels and this is equal to cos square phi plus sine square phi which is equal to 1. Since we know that cos square theta plus sine square theta is equal to 1 and this is equal to the RHS. Thus, we get n square into n square into n square plus n square is equal to 1. So, hence proved this can be a key session. Hope you have understood the solution of this question.