If-a-1-a-2-a-3-are-in-AP-and-d-is-the-common-difference-then-tan-1-d-1-a-1-a-2-tan-1-d-1-a-2-a-3-

Question Number 130581 by bramlexs22 last updated on 27/Jan/21

If a_{1}, a_{2}, a_{3} are in AP and d is the common

difference then \tan^{- 1} (\frac{d}{1 + a_{1} a_{2}}) + \tan^{- 1} (\frac{d}{1 + a_{2} a_{3}}) = ?

Answered by som(math1967) last updated on 27/Jan/21

\tan^{- 1} (\frac{a_{2} - a_{1}}{1 + a_{1} a_{2}}) + \tan^{- 1} (\frac{a_{3} - a_{2}}{1 + a_{2} - a_{3}})

= \tan^{- 1} a_{2} - \tan^{- 1} a_{1} + \tan^{- 1} a_{3} - \tan^{- 1} a_{2}

= \tan^{- 1} a_{3} - \tan^{- 1} a_{1}

[d = a_{2} - a_{1} = a_{3} - a_{2}]

Commented by bramlexs22 last updated on 27/Jan/21

Commented by bramlexs22 last updated on 27/Jan/21

= \tan^{- 1} (\frac{a_{3} - a_{1}}{1 + a_{1} a_{3}}) = \tan^{- 1} (\frac{2 d}{1 + a_{1} a_{3}})