$$\cos(2x)-\cos(3x)=0$$
I am trying to solve this equation but get stuck every time in the middle of the exercise. Can somebody help me please?
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Sign up to join this community$$\cos(2x)-\cos(3x)=0$$
I am trying to solve this equation but get stuck every time in the middle of the exercise. Can somebody help me please?
Notice, we have $$\cos(3x)=\cos(2x)$$ $$3x=2n\pi\pm 2x$$ Where, $n$ is any integer
Now, we have the following solutions
$$3x=2n\pi+2x\implies \color{red}{x=2n\pi}$$ or $$3x=2n\pi-2x\implies \color{red}{x=\frac{2n\pi}{5}}$$
Edit -1 (Thanks @Bernard)
Observe the second set of solutions contains the first.
Hint: $$\cos 2x - \cos 3x = 2 \sin \frac{5x}{2}\sin\frac{x}{2}$$
use that $$\cos(x)-\cos(y)=-2 \sin \left(\frac{x}{2}-\frac{y}{2}\right) \sin \left(\frac{x}{2}+\frac{y}{2}\right)$$