Choose the correct answer from the given four options.The value o… - Vidyadip

MCQ

Choose the correct answer from the given four options.The value of $\sin\big[2\tan^{-1}(0.75)\big]$ is equal to:

A
$0.75$
B
$1.5$
✓
$0.96$
D
$\sin1.5$

✓

Answer

Correct option: C.

$0.96$

We have, $\sin\big[2\tan^{-1}(0.75)\big]$
$=\sin\Big(2\tan^{-1}\frac{3}{4}\Big)$
$=\sin\Bigg(\sin^{-1}\frac{2.\frac{3}{4}}{1+\frac{9}{16}}\Bigg)$
$\Big(\because\ 2\tan^{-1}\text{x}=\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}\Big)$
$=\sin\Bigg(\sin^{-1}\frac{\frac{3}{2}}{\frac{25}{16}}\Bigg)$
$=\sin\Big(\sin^{-1}\frac{24}{25}\Big)=\frac{24}{25}=0.96$

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