 Hello and welcome to this session. Let us understand the following problem today. Using properties of determinants, prove that determinant sin alpha cos alpha cos of alpha plus delta sin beta cos beta cos of beta plus delta sin gamma cos gamma cos of gamma plus delta is equal to 0. Now let us write this solution. Consider LHS which is equal to the determinant sin alpha cos alpha cos of alpha plus delta sin beta cos beta cos of beta plus delta sin gamma cos gamma cos of gamma plus delta. Now applying the formula cos A plus B here in this column we get sin alpha cos alpha cos alpha cos delta minus sin alpha sin delta and here we get sin beta cos beta cos beta cos delta minus sin beta sin delta sin gamma cos gamma cos delta cos gamma minus sin gamma sin delta. Now applying C3 tends to C3 plus sin delta of C1 minus cos delta into C2 which is equal to sin alpha cos alpha as it is. Now C3 tends to applying to this column C3 first of all this as it is cos alpha cos delta minus sin alpha sin delta now plus sin delta of into C1 so C1 is this so we get here plus sin delta sin alpha minus cos delta C2 now the second row sin beta cos beta as it is now this as it is we get here cos beta cos alpha minus sin beta sin delta now plus sin delta into C1 which is sin beta sin beta minus cos delta C2 that is cos beta now the third row sin gamma cos gamma this as it is so cos gamma cos delta minus sin gamma sin delta plus sin delta C1 that is sin gamma minus cos delta into C2 that is cos gamma now we see here this and this gets cancels and this and this gets cancels similarly here this and this gets cancels cancels and this and this gets cancels. Similarly this and this gets cancels and this and this gets cancels. So we get which is equal to sin alpha cos alpha 0 sin beta cos beta 0 sin gamma cos gamma 0. Now since column C3 has all the elements as 0 so the determinant is equal to 0 which is equal to RHS therefore LHS is equal to RHS hence proved. I hope you understood the problem. Bye and have a nice day.