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Question

Solve for $$x$$ : 
$$2\tan^{-1}(\cos x)=\tan^{-1}(2 cosec x) $$


Solution

$$2\tan ^{ -1 } (\cos  x)=\tan ^{ -1 } (2cosecx)$$
$$\displaystyle \tan ^{ -1 }{ \left( \frac { 2\cos { x }  }{ 1-\cos ^{ 2 }{ x }  }  \right)  } =\tan ^{ -1 } (2cosecx)$$
$$\displaystyle \frac { \cos { x }  }{ \sin ^{ 2 }{ x }  } =cosecx$$
$$cosecx(\cot { x } -1)=0$$
$$\cot { x } =1\quad (\because cosecx\ne 0)$$
$$\displaystyle x=n\pi +\frac { \pi  }{ 4 } ,n\in Z$$

Mathematics

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