First slide

Properties of ITF

Question

If $a > b > c > 0,$ then $\cot^{- 1} (\frac{a b + 1}{a - b}) + \cot^{- 1} (\frac{b c + 1}{b - c}) + \cot^{- 1} (\frac{c a - 1}{c - a})$

Moderate

A

0
B

$\frac{π}{2}$
C

$π$
D

$\frac{3 π}{2}$

Solution

$G i v e n a > b > c > 0 . N o w c o t^{- 1} (\frac{a b + 1}{a - b}) + c o t^{- 1} (\frac{b c + 1}{b - c}) + c o t^{- 1} (\frac{c a + 1}{c - a}) = c o t^{- 1} (\frac{a b + 1}{a - b}) + c o t^{- 1} (\frac{b c + 1}{b - c}) + c o t^{- 1} (- (\frac{a c + 1}{a - c})) = c o t^{- 1} a - c o t^{- 1} b + c o t^{- 1} b - c o t^{- 1} c + π - (\cot^{- 1} a - \cot^{- 1} c) (∵ if x > y, then \cot^{- 1} (\frac{xy + 1}{x - y}) = \cot^{- 1} x - \cot^{- 1} y, also \cot^{- 1} (- x) = π - \cot^{- 1} x) = π$

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