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Question: If \(2 \tan ^ { - 1 } ( \cos x ) = \tan ^ { - 1 } ( 2 \operatorname { cosec } x )\) then x =...

If 2tan⁡−1(cos⁡x)=tan⁡−1(2cosec⁡x)2 \tan ^ { - 1 } ( \cos x ) = \tan ^ { - 1 } ( 2 \operatorname { cosec } x ) then x =

A

3π4\frac { 3 \pi } { 4 }

B

π4\frac { \pi } { 4 }

C

π3\frac { \pi } { 3 }

D

None of these

Answer

π4\frac { \pi } { 4 }

Explanation

Solution

We have 2tan⁡−1(cos⁡x)=tan⁡−1(2cosec⁡x)\tan ^ { - 1 } ( \cos x ) = \tan ^ { - 1 } ( 2 \operatorname { cosec } x )

⇒ tan⁡−1(2cos⁡x1−cos⁡2x)\tan ^ { - 1 } \left( \frac { 2 \cos x } { 1 - \cos ^ { 2 } x } \right) = tan⁡−1(2cosec⁡x)\tan ^ { - 1 } ( 2 \operatorname { cosec } x )

sin⁡x=cos⁡x\sin x = \cos x

⇒x=π4\Rightarrow x = \frac { \pi } { 4 }.