First slide

Properties of ITF

Question

If $\tan^{- 1} y = 4 \tan^{- 1} x$ , then 1/y is zero for

Moderate

A

$x = 1 \pm \sqrt{2}$
B

$x = \sqrt{2} \pm \sqrt{3}$
C

$x = 3 \pm 2 \sqrt{2}$
D

all values of x

Solution

If we put x = tanθ, the given equality becomes tan^–1y = 4θ
$\Rightarrow y = \tan 4 θ = \frac{2 \tan 2 θ}{1 - \tan^{2} 2 θ} = \frac{2 [\frac{2 \tan θ}{1 - \tan^{2} θ}]}{1 - {(\frac{2 \tan θ}{1 - \tan^{2} θ})}^{2}}$
$= \frac{2 \times 2 x (1 - x^{2})}{{(1 - x^{2})}^{2} - 4 x^{2}} = \frac{4 x (1 - x^{2})}{1 - 6 x^{2} + x^{4}}$
$∴ \frac{1}{y} is zero if x^{4} - 6 x^{2} + 1 = 0$
$\Rightarrow x^{2} = \frac{6 \pm \sqrt{36 - 4}}{2} = 3 \pm 2 \sqrt{2} = (1 \pm \sqrt{2})^{2}$

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