If `0lta_1lta_2lt....lta_n ,` then prove that `tan^(-1)((a_1x-y) /(x+a_1y))+tan^(-1)((a_2-a_1) /(1+a_2a_1))+tan^(-1)((a_3-a_2)/(1+a_3a_2))+.......+tan^(-1)((a_n-a_(n-1)) /(1+a_n a_(n-1)))+tan^(-1)(1/(a_n))=tan^(-1)(x/y)dot`

Prove that: tan^-1 ((a_1x-y)/(x+a_1y))+tan^-1 ((a_2-a_1)/(1+a_1a_1))+tan^-1 ((a_3-a_2)/(1+a_3a_2))+…+tan^-1, ((a_n-a_(n-1)/(1+a_na_(n-1))+tan^-1 (1/a_n)=tan^-1, x/y

If c_i >0 for i=1,\ 2,\ ,\ n , prove that tan^(-1)((c_1x-y)/(c_1y+x))+tan^(-1)((c_2-c_1)/(1+c_2c_1))+tan^(-1)((c_3-c_2)/(1+c_3c_2))+\ dot+tan^(-1)(1/(c_n))=tan^(-1)(x/y)

Prove that tan ^ (- 1) ((1) / (3)) + tan ^ (- 1) ((1) / (7)) + tan ^ (- 1) ((1) / (13)) + ......... + tan ^ (- 1) ((1) / (n ^ (2) + n + 1)) +

Given that: (a_(1)+ib_(1))(a_(2)+ib_(2)) . . . (a_(n)+ib_(n))=c+id , show that: tan^(-1)((b_(1))/(a_(1)))+tan^(-1)((b_(2))/(a_(2)))+ . . .+tan^(-1)((b_(n))/(a_(n))) =mpi+tan^(-1)((d)/(c)) .

If a_(1),a_(2),a_(3),a_(n) is an A.P.with common difference d, then prove that tan[tan^(-1)((d)/(1+a_(1)a_(2)))+tan^(-1)((d)/(1+a_(2)a_(3)))+tan^(-1)((d)/(1+a_(n-1)a_(n)))]=((n-1)d)/(1+a_(1)a_(n))

Given that (a_(1)+ib_(1))(a_(2)+ib_(2)) . . . .(a_(n)+ib_(n))=c+id , show that: tan^(-1)((b_(1))/(a_(1)))+tan^(-1)((b_(2))/(a_(2)))+ . . .+tan^(-1)((b_(n))/(a_(n)))=mpi+tan^(-1)((d)/(c)) .

If a_(1),a_(2),a_(3),….a_(n) is a.p with common difference d then tan{tan^(-1)((d)/(1+a_(1)a_(2)))+tan^(-1)((d)/(1+a_(2)a_(3))) +..+ tan^(-1)((d)/(1+a_(n-1)a_(n)))} is equal to

If a_(1), a_(2), a_(3),...., a_(n) is an A.P. with common difference d, then prove that tan[tan^(-1) ((d)/(1 + a_(1) a_(2))) + tan^(-1) ((d)/(1 + a_(2) a_(3))) + ...+ tan^(-1) ((d)/(1 + a_( - 1)a_(n)))] = ((n -1)d)/(1 + a_(1) a_(n))

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""