Evaluate: ( ^ -1 x+ ^ -1 1 x ) for x>0 - Vidyadip

Question

Evaluate:
$\sin\Big(\tan^{-1}\text{x}+\tan^{-1}\frac{1}{\text{x}}\Big)\text{ for }\text{x}>0$

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Answer

$\sin\Big(\tan^{-1}\text{x}+\tan^{-1}\frac{1}{\text{x}}\Big)$
$\sin\Big(\tan^{-1}\text{x}+\cot^{-1}{\text{x}}\Big)$ $\Big[\because\ \tan^{-1}\text{x}=\cot^{-1}\frac{1}{\text{x}}\Big]$
$=\sin\Big(\frac{\pi}{2}\Big)$
$=1$

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