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Question

If $$2 \tan ^{-1}(\cos \, x)=\tan^{-1}(2 cosec\, x)$$, then the value of x is.


A
3π4
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B
π4
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C
π3
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D
None of these
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Solution

The correct option is B $$\dfrac{ \pi}{4}$$
$$2\tan^{-1}(\cos x)=\tan^{-1}(2 \, cosec \, x)$$

$$\Rightarrow \tan ^{-1}\left (\dfrac{2 cos \, x}{1-cos^2 x}  \right )=\tan^{-1}(2 cosec \, x)$$  .... $$\left[ \because 2\tan^{-1} x=\tan^{-1}(\dfrac{2x}{1-x^2})\right]$$

$$\Rightarrow \dfrac{2 \cos \, x}{1-\cos^2 x}=2 \ cosec x$$

$$\Rightarrow \dfrac{2 \cos \, x}{\sin^2 x}=2 cosec x $$

$$\Rightarrow  \sin\ x= \cos\ x\Rightarrow x=\dfrac{\pi}{4}$$
Hence, option B is correct.

Mathematics

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