Question

If $\cos 2x-3\cos x+1=\frac{\csc x}{\cot x-\cot 2x}$, then which of the following is true?

• A. $x=(2n+1)\frac{\pi }{2},n\in I$
• B. $x=2n\pi ,n\in I$
• C. $x=2n\pi \pm {\cos }^{-1}\left(\frac{2}{5}\right),n\in I$
• D. None

Answer 1

The correct option is D None

$\cos 2x-3\cos x+1=\frac{\csc x}{\cot x-\cot 2x}\Rightarrow \frac{1}{\cot x-\cot 2x}(-\cot x+3\cos x\cot x-\cos 2x\cot x+\cot 2x-3\cos x\cot 2x+\cos 2x\cot 2x+\csc x)=0\Rightarrow -2\sin (\frac{\pi }{4}-\frac{x}{2})\sin (\frac{\pi }{4}+\frac{x}{2})(2\cos x-5)=0$
$\Rightarrow \cos x=\frac{5}{2}$ or $\sin (\frac{\pi }{4}-\frac{x}{2})=0$ or $\sin (\frac{\pi }{4}+\frac{x}{2})=0$
$\Rightarrow x=\frac{\pi }{2}-2n\pi ,x=2m\pi +\frac{\pi }{2}$