f(x)=cos(x)+(1/2)cos(2x)
Show that f(-x)=f(x)
Is the fact that cos(-a)=cos(a) proof enough?

Question

f(x)=cos(x)+(1/2)cos(2x)
Show that f(-x)=f(x)
Is the fact that cos(-a)=cos(a) proof enough?

thomas5267 · Answer

I think you still have to show that $f(x)=f(-x)$.

deoxna · Answer

Ok so I write out: 
$$f(-x)=\cos(-x)+(1/2)\cos(-2x)$$
then 
$$\cos(-a)=\cos(a)$$
$$f(-x)=\cos(x)+(1/2)\cos(2x)$$

I just can't see any other way of proving this, other than perhaps double angle identities, but then I would still have to rely on cos(x)=cos(-x) to prove the first term.

thomas5267 · Answer

I think using $\cos (x)=\cos (-a)$ will do.

deoxna · Answer

Ok, thank you. I wasn't sure about that one...

thomas5267 · Answer

I think you can't use double angle identity to prove it though.

deoxna · Answer

Yeah, cause you'd still have (cos(2x)^2 which you can't really simplify...