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Relation between Inverses of Trigonometric Functions and Their Reciprocal Functions

The number of...

Question

The number of positive integral solutions of the equation $t a n^{- 1} x + c o t^{- 1} y = t a n^{- 1} 3$ is :

0

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Solution

The correct option is A $2$
$c o t^{- 1} y = t a n^{- 1} 3 - t a n^{- 1} x$
$c o t^{- 1} y = t a n^{- 1} (\frac{3 - x}{1 + 3 x})$
Take tan on both sides
$\frac{1}{y} = \frac{3 - x}{1 + 3 x}$
$y = \frac{1 + 3 x}{3 - x} \Rightarrow y > 0$
Put $x = 0, y = \frac{1}{3}$
$\frac{1 + 3 x}{3 - x} > 0$
$x = 1, y = 2$
$\frac{3 x + 1}{x = 3} < 0$
$x = 2, y = 7$
$x ϵ (- \frac{1}{3}, 3)$
The integral values of $x = 0, 1, 2$

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