[ ^ -1 ( ^ -1 x ) ] is equal to: - Vidyadip

MCQ

$\sin\big[\cot^{-1}\big\{\tan\big(\cos^{-1}\text{x}\big)\big\}\big]$ is equal to:

✓
$\text{x}$
B
$\sqrt{1-\text{x}^2}$
C
$\frac{1}{\text{x}}$
D
None of these

✓

Answer

Correct option: A.

$\text{x}$

Put $\cos^{-1}\text{x}=\text{u}$
$\sin\big[\cot^{-1}\big\{\tan\big(\cos^{-1}\text{x}\big)\big\}\big]$
$=\sin\big[\cot^{-1}\{\tan(\text{u})\}\big]$
$=\sin\Big[\cot^{-1}\Big\{\cot\Big(\frac{\pi}{2}-\text{u}\Big )\Big\}\Big]$
$=\sin\Big[\frac{\pi}{2}-\text{u}\Big]$
$=\cos\text{u}$
$=\text{x} \big(\therefore \cos^{-1}\text{x}=\text{u}\Rightarrow\text{x}=\cos\text{u}\big)$

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