First slide

Properties of ITF

Question

The positive integral solution of $\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^{2}}} = \sin^{- 1} \frac{3}{\sqrt{10}} is$

Easy

A

x = 1, y = 2; x = 2, y = 7
B

x = 1, y = 3; x = 2, y = 4
C

x = 0, y = 0; x = 3, y = 4
D

none of these

Solution

Converting cos and sin into tan, we have,
$\begin{array}{l} \tan^{- 1} x + \tan^{- 1} (\frac{1}{y}) = \tan^{- 1} (\frac{3}{1}) \\ \Rightarrow \tan^{- 1} [\frac{x + (\frac{1}{y})}{1 - x (\frac{1}{y})}] = \tan^{- 1} (\frac{3}{1}) \Rightarrow \frac{xy + 1}{y - x} = 3 \end{array}$
∴ The equality is true for
x = 1, y = 2 and for x = 2, y = 7

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