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  • <img alt="{\cos ^{ - 1}}[\cos \;(2{\cot ^{ - 1}}\;(\sqrt 2 - 1)]=" src="http://entrancecorner.oncodecogs.com/gif.latex?%7B%5Ccos%20%5E%7B%20-%201%7D%7D%5B%5Ccos%20%5C%3B%282%7B%5Ccot%20%5E%7B%20-%2

{\cos ^{ - 1}}[\cos \;(2{\cot ^{ - 1}}\;(\sqrt 2 - 1)]=

Option: 1

\sqrt 2 - 1


Option: 2

1 - \sqrt 2


Option: 3

\frac{\pi }{4}


Option: 4

\frac{3\pi }{4}


Answers (1)

best_answer

Let  {\cot ^{ - 1}}(\sqrt 2 - 1) = \theta      \Rightarrow   (\sqrt 2 - 1) = \cot \theta

Therefore

{\cos ^{ - 1}}\;[\cos 2\theta ] = {\cos ^{ - 1}}\;\left[ {\frac{{1 - {{\tan }^2}\theta }}{{1 + {{\tan }^2}\theta }}} \right]  

 ={\cos ^{ - 1}}\;\left[ {\frac{{{{\cot }^2}\theta - 1}}{{{{\cot }^2}\theta + 1}}} \right]\; = \;{\cos ^{ - 1}}\;\left[ {\frac{{2 - 2\sqrt 2 }}{{4 - 2\sqrt 2 }}} \right]

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

Posted by

Suraj Bhandari

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