Question

Prove the identity: cos(3x) + cos(x) = 2cos(2x)cos(x)

Answer 1

$cos \alpha +cos \beta =2\cdot cos \frac{ \alpha + \beta }{2} \cdot cos \frac{ \alpha - \beta }{2}\\-----------------\\\\ \alpha =3x\ \ \ and\ \ \ \beta =x$

$L=cos \alpha +cos \beta =cos(3x)+cos(x)\\\\\Rightarrow\ \ \ 2\cdot cos \frac{ \alpha + \beta }{2} \cdot cos \frac{ \alpha - \beta }{2}=2\cdot cos \frac{3x+x}{2} \cdot cos \frac{3x-x}{2} =\\\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =2\cdot cos \frac{4x}{2} \cdot cos \frac{2x}{2} =\\\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =2\cdot cos(2x)\cdot cos (x)=R$