If a₁, a₂,..... a_n is an AP with common difference d, then tan [tan^-1 d / [1 + a₁a₂] + tan^-1 d / [1 + a₂a₃] + …. + tan^-1 d / [1 + a_n-1a_n]] =

1) [n – 1] d / [a₁ + a_n]

2) [n – 1] d / [1 + a₁a_n]

3) nd / [1 + a₁a_n]

4) [a_n – a₁] / [a_n + a₁]

Solution: (2) [n – 1] d / [1 + a₁a_n]

tan^-1 d / [1 + a₁a₂] + tan^-1 d / [1 + a₂a₃] + …. Tan^-1 d / [1 + a_n-1a_n]

= tan^-1 [a₂ – a₁] / [1 + a₁a₂] + tan^-1 [a₃ – a₂] / [1 + a₂a₃] + …. + tan^-1 [a_n – a_n-1] / [1 + a_n-1a_n]

= tan^-1 a₂ – tan^-1 a₁ + tan^-1 a₃ – tan^-1 a₂ + …. + tan^-1 a_n – tan^-1 a_n-1 +

= tan^-1 a_n – tan^-1 a₁

= tan^-1 [a_n – a₁] / [1 + a_na₁]

= tan^-1 [(n – 1) d] / [1 + a₁a_n]

= tan[tan^-1 [(n – 1) d] / [1 + a₁a_n]]

= [(n – 1) d] / [1 + a₁a_n]