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?

Answer 1

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

Answer 2

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.

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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