Find all solutions in the interval [0, 2π).
-$\cos ^ { 2 } \left( \frac { x } { 2 } \right) = \cos ^ { 2 } x$
A) $0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$
B) $0 , \frac { 2 \pi } { 3 }$
C) $\frac { \pi } { 6 } , \pi , \frac { 11 \pi } { 6 }$
D) $0 , \frac { \pi } { 6 }$

Question

Quizplus · Accepted Answer

The equation $\cos^2\left(\frac{x}{2}\right) = \cos^2x$ can be solved by recognizing that for the two expressions to be equal, either $\cos\left(\frac{x}{2}\right) = \cos(x)$ or $\cos\left(\frac{x}{2}\right) = -\cos(x)$. Solving these equations within the given interval [0, 2π) leads to the solutions $x = 0$, $x = \frac{2\pi}{3}$, and $x = \frac{4\pi}{3}$, which are the points where the original equation holds true.

Find All Solutions in the Interval [0, 2π) cos⁡2(x2)=cos⁡2x\cos ^ { 2 } \left( \frac { x } { 2 } \right) = \cos ^ { 2 } xcos2(2x​)=cos2x

Find All Solutions in the Interval [0, 2π) $\cos ^ { 2 } \left( \frac { x } { 2 } \right) = \cos ^ { 2 } x$