If (a1,a2,a3......an) are in A>P> with common difference d. then the sum of the series: sin d [cosec a1.cosec a2 + cosec a2.cosec a3 +........+ cosec a(n-1).cosec an ] is what ??

Given: a₁, a₂, a₃, ..... a_n are in A.P. with common difference d

Then d = (a₂ – a₁) = (a₃ – a₂) ......  (a_n – a_n-1)

Now,

sin d (cosec a₁ cosec a₂ + cosec a₂ cosec a₃ + ..... cosec a_n-1 cosec a_n)

= cot a₁ – cot a_n