Question

${\cos }^{-1}\left(\cos \left(2{\cot }^{-1}(\sqrt{2}-1)\right)\right)$ is equal to

• A. $\sqrt{2}-1$
• B. $\frac{\pi }{4}$
• C. $\frac{3\pi }{4}$
• D. None of these

Answer 1

Answer: (C)

${\cos }^{-1}\left(\cos \left(2{\cot }^{-1}(\sqrt{2}-1)\right)\right)$
$={\cos }^{-1}\left(\cos \left(2{\cos }^{-1}\left(\frac{\sqrt{2}-1}{\sqrt{4}-2\sqrt{2}}\right)\right)\right)$
$={\cos }^{-1}\left(\cos \left({\cos }^{-1}\left(\frac{2(\sqrt{2}-1{)}^2}{4-2\sqrt{2}}-1\right)\right)\right)$
$={\cos }^{-1}\left(\cos \left({\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)\right)\right)$
$={\cos }^{-1}\left(\cos \left(\pi -{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)\right)$
$={\cos }^{-1}\left(-\cos \left({\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)\right)$
$={\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)=\pi -\frac{\pi }{4}=\frac{3\pi }{4}$