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  4. cos -1( cos (2 cot -1(√2-1))) is equal to

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

Inverse Trigonometric Functions

Solution:

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