Question

The general solution of the equation $2\cos 2x=3\cdot 2{\cos }^{2}x-4$ is

• A. $x=2n\pi ,nϵI$
• B. $x=n\pi ,nϵI$
• C. $x=n\frac{\pi }{4},nϵI$
• D. $x=n\frac{\pi }{2},nϵI$

Answer 1

Answer: (A)

$x=2n\pi ,nϵI$

Given,

$2\cos \left(2x\right)=3\cdot 2{\cos }^2\left(x\right)-4$

$-6{\cos }^2\left(x\right)+2\cos \left(2x\right)+4=0$

$4+(-1+2{\cos }^2\left(x\right))\cdot 2-6{\cos }^2\left(x\right)=0$

upon simplification, we get,

$2-2{\cos }^2\left(x\right)=0$

$\Rightarrow \cos \left(x\right)=1,\cos \left(x\right)=-1$

$\cos \left(x\right)=1:x=2\pi n$

and $\cos \left(x\right)=-1:x=\pi +2\pi n$

$x=2\pi n,x=\pi +2\pi n$