Question

transformati in produs cosx+cos2x+cos3x+cos4x

Answer 1

[tex]\text{Folosim identitatea:}\cos a+\cos b=2\cos \dfrac{a+b}{2}\cos \dfrac{a-b}{2}\\ \cos x+\cos 2x+\cos 3x+\cos 4x=(\cos 4x+\cos x)+(\cos 3x+\cos 2x)=\\ =2\cos \dfrac{4x+x}{2}\cdot \cos \dfrac{4x-x}{2}+2\cos \dfrac{3x+2x}{2}\cos \dfrac{3x-2x}{2}= \\ =2\cos \dfrac{5x}{2}\cos\dfrac{3x}{2}+2\cos \dfrac{5x}{2}\cos \dfrac{x}{2}= \\ =2\cos \dfrac{5x}{2}\left(\cos \dfrac{3x}{2}+\cos \dfrac{x}{2}\right)\\ \text{In mod normal ne putem opri aici,dar daca vrei sa continuam:}[/tex]
$=4\cos \dfrac{5x}{2}\cos\dfrac{\frac{3x}{2}+\frac{x}{2}}{2}\cos \dfrac{\frac{3x}{2}-\frac{x}{2}}{2}=\boxed{4\cos \dfrac{5x}{2}\cdot \cos x\cdot \cos \dfrac{x}{2}}$