Question

Calcula «x» si: Cos5x + Cosx = Cos2x

Answer 1

Utilizaremos la siguiente identidad

$\cos\alpha+\cos\beta=\dfrac{1}{2}\cos\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha-\beta}{2}\right)$

Resolución

$\cos (5x)+\cos(x)=\cos(2x)\\\\\dfrac{1}{2}\cos(3x)\cos(2x)=\cos(2x)\\\\\dfrac{1}{2}\cos(3x)\cos(2x)-\cos(2x)=0\\\\\cos(2x)\left[\dfrac{1}{2}\cos(3x)-1\right]=0\\\\\cos(2x)=0\vee \cos(3x)=2\Longrightarrow \cos(2x)=0 \\ \\--------~~\text{Puesto que \cos \theta\in[-1,1] }\\ \\2x\in\left\{\dfrac{2k-1}{2}\pi: k\in\mathbb Z\right\}\\\\\\x\in\left\{\dfrac{2k-1}{4}\pi: k\in\mathbb Z\right\}$