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Question

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


Solution

$$\displaystyle 2{ \tan }^{ -1 }\left( \cos { x }  \right) ={ \tan }^{ -1 }\left( 2co \sec x \right) $$

$$\displaystyle \Rightarrow { \tan }^{ -1 }\left( \frac { 2\cos { x }  }{ 1-{ \cos }^{ 2 }x }  \right) ={ \tan }^{ -1 }\left( 2co\sec x \right)$$

$$\displaystyle \Rightarrow \frac { 2\cos { x }  }{ 1-{ \cos }^{ 2 }x } =2cosec x$$

$$\displaystyle \Rightarrow \frac { 2\cos { x }  }{ { \sin }^{ 2 }x } =\frac { 2 }{ \sin { x }  } $$

$$\displaystyle \Rightarrow \cos { x } =\sin { x } $$

$$\displaystyle \Rightarrow \tan { x } =1$$

$$\displaystyle \therefore x=\frac { \pi  }{ 4 } $$

Mathematics
RS Agarwal
Standard XII

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