If `a_1, a_2,a_3,..........., a_n` are in A.P. with common difference d, prove that :
`sin d [cosec a_1 coseca_2+ coseca_2coseca_3+....coseca_(n-1)coseca_n]= cot a_1-a_n`.

If a_1, a_2 , ............, a_n are in A.P.with common difference d ne 0 , then [sin d] [sec a_1 sec a_2 + sec a_2sec a_3+............+ sec a_(n-1) sec a_n] is equal to :

If a_1, a_2,a_3,..........., a_n be an A.P. of non-zero terms, prove that : 1/(a_1 a_2)+1/(a_2 a_3)+ ............. + 1/(a_(n-1) a_n)= (n-1)/(a_1 a_n) .

If a_1, a_2 , ............, a_n are in A.P. and a_i >0 for all i , prove that : 1/(sqrt a_1+sqrt a_2)+1/(sqrt a_2+sqrt a_3)+.............+ 1/(sqrt (a_(n-1))+sqrt a_n)= (n-1)/(sqrt a_1+sqrt a_n) .

If theta_(1),theta_(2),theta_(2), theta_(3),"…………" Theta_(n) are in AP whose common difference is d, show that sec theta_(1) sec theta_(2)+ sec_(3) +" …" + sec theta _(n-1)sec theta_(n)=(tantheta_(n)-tan theta_(1))/(sin d )

Prove that (cosecA)/(cosecA-1)+(cosecA)/(cosecA+1)=2sec^2A .

If a_1,a_2,a_3,.....,a_(n+1) be (n+1) different prime numbers, then the number of different factors (other than1) of a_1^m.a_2.a_3...a_(n+1) , is

For the A.P., a_1, a_2,a_3, ............., if a_4/a_7= 2/3 , find a_6/a_8 .

If the sequence a_1, a_2, a_3,....... a_n ,dot forms an A.P., then prove that a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)

Prove that 1/(cosecA+cotA)=(cosecA-cotA) .

A person is to count 4500 currency notes, Let a_n denote the number of notes he courts in the nth minute. If a_1=a_2= ........ = a_10 =150 and a_10, a_11 , ............ are in an A.P. with common difference -2, then the time taken by him to count all notes is :