If 2 tan^-1(cos x)=tan^-1(2 cosec x) , then the value of x is

Correct answer: (B) (π)/4. Given, 2 tan^-1(cos x)=tan^-1(2 cosec x) ∴ tan^-1 ((2 cos x)/(1-cos^2 x) )=tan^-1(2 cosec x) ⇒ (2 cos x)/(1-cos^2 x)=2 cosec x ⇒ sin x=cos x ⇒ x=(π)/4

Q61 / 170 · Manipal (MET) 2016 · Mathematics

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

A
$\frac{3 \pi}{4}$
Correct answer
B
$\frac{\pi}{4}$
C
$\frac{\pi}{3}$
D
None of these

Solution

$\begin{aligned} & \text { Given, } 2 \tan ^{-1}(\cos x)=\tan ^{-1}(2 \operatorname{cosec} x) \\ & \therefore \tan ^{-1}\left(\frac{2 \cos x}{1-\cos ^2 x}\right)=\tan ^{-1}(2 \operatorname{cosec} x)\end{aligned}$ $\begin{aligned} & \Rightarrow \quad \frac{2 \cos x}{1-\cos ^2 x}=2 \operatorname{cosec} x \\ & \Rightarrow \sin x=\cos x \Rightarrow x=\frac{\pi}{4}\end{aligned}$

Hence, the correct answer is option: 'B'.

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

Solution

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