Question

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

• A. $\frac{3\mathrm{\pi }}{4}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. None of these

Answer 1

The correct option is B

$\frac{\mathrm{\pi }}{4}$

Explanation for the correct option:

Find the value of $x$:

Given,

$2{\tan }^{-1}(\cos x)={\tan }^{-1}(2\cos ecx)$

$\Rightarrow$${\tan }^{-1}\left(\frac{2\cos x}{1–{\cos }^{2}x}\right)={\tan }^{-1}(2\cos ecx)$ $\left[\mathbf{\because }\mathbf{2}\mathbf{}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{A}\mathbf{}\mathbf{=}\mathbf{}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\left(\frac{2A}{1-A^2}\right)\right]$

$\Rightarrow$ $\frac{2\cos x}{1–{\cos }^{2}x}=2\cos ecx$

$\Rightarrow$ $\frac{2\cos x}{{\sin }^{2}x}=2\cos ecx$ $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

$\Rightarrow$ $\cos x=\sin x$

$\mathbf{\therefore }\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{4}}$

Hence, Option ‘B’ is Correct.