Question

If $2ta{n}^{-1}(cosx)=ta{n}^{-1}(2cosecx)$ , then $x$ is equal to

• A. $\pi /2$
• B. $\pi /3$
• C. $\pi /4$
• D. $\pi /6$

Answer 1

Answer: (C)

Given, $2ta{n}^{-1}\left(cosx\right)=ta{n}^{-1}\left(2cosecx\right)$
$\Rightarrow 2ta{n}^{-1}\left(cosx\right)=ta{n}^{-1}\left(2cosecx\right)$
$\Rightarrow ta{n}^{-1}\left(\frac{2cosx}{1-co{s}^{2}x}\right)=ta{n}^{-1}\left(2cosecx\right)$
$\Rightarrow \frac{2cosx}{si{n}^{2}x}=\frac{2}{sinx}$
$\Rightarrow si{n}^{2}x-sinx\cdot cosx=0$
$\Rightarrow sinx\left(sinx-cosx\right)=0$
$\Rightarrow sinx=0$ or $tanx=1$
$\Rightarrow x=0$ or $x=\pi /4$
Here, $x=\pi /4$ satisfies the given equation
So, $x\ne 0$