First slide

Properties of ITF

Question

The number of solutions to the equation $\tan^{- 1} (\frac{x}{3}) + \tan^{- 1} (\frac{x}{2}) = \tan^{- 1} x$ is

Easy

A

3
B

2
C

1
D

0

Solution

$\tan^{- 1} (\frac{x}{3}) + \tan^{- 1} (\frac{x}{2}) = \tan^{- 1} x$
Or $\tan^{- 1} (\frac{x / 3 + x / 2}{1 - x^{2} / 6}) = \tan^{- 1} x$
$\Rightarrow (\frac{5 x}{6 - x^{2}}) = x$
$\begin{array}{l} \Rightarrow x = 0 or x^{2} - 1 = 0 \Rightarrow x = \pm 1 \\ T h e r e f o r e, x = {- 1, 0, 1} \Rightarrow 3 solutions \end{array}$

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