First slide

Properties of ITF

Question

If $x = \sin (2 \tan^{- 1} 2)$ and $y = \sin [\frac{1}{2} \tan^{- 1} \frac{4}{3}]$ then

Easy

A

$x = y$
B

$x > y$
C

$x < y$
D

$\frac{2}{\sqrt{5}}$

Solution

Put $\tan^{- 1} 2 = A$ and $\tan^{- 1} \frac{4}{3} = B$ .
$\Rightarrow \tan A = 2 a n d \tan B = \frac{4}{3} \Rightarrow \sin A = \frac{2}{\sqrt{5}}, \cos A = \frac{1}{\sqrt{5}} a n d \cos B = \frac{3}{5} ∴ x = \sin (2 \tan^{- 1} 2) = \sin 2 A = 2 \sin A \cos A = 2 \frac{2}{\sqrt{5}} \frac{1}{\sqrt{5}} = \frac{4}{5} a n d y = \sin [\frac{1}{2} \tan^{- 1} \frac{4}{5}] = \sin \frac{B}{2} = \sqrt{\frac{1 - \cos B}{2}} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \frac{1}{\sqrt{5}} \Rightarrow x > y$

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