What is the set $\text{Span}(\sin(x),\cos(x))\cap \mathbb{R}$? Find $\text{Span}(\sin(x),\cos(x))\cap \mathbb{R}.$
I know that $\text{Span}(\sin x,\cos x)=\{a\sin x+b\cos x:a,b\in \mathbb{R}\}.$ Now $a\sin x+b\cos x$ is a real number and so $\text{Span}(\sin x,\cos x)\subset \mathbb{R}.$ Thus we have that $\text{Span}(\sin(x),\cos(x))\cap \mathbb{R}=\text{Span}(\sin x,\cos x).$ But the book says that the $$\text{Span}(\sin(x),\cos(x))\cap \mathbb{R}=\{0\}.$$ Why?

Suppose $a\sin x+b\cos x$ is a constant $c$. Plugging in $x=0,\pi$ we get respectively $a\cdot 0+b\cdot 1=c$ and $a\cdot 0+b\cdot(-1)=c$, i.e. $b=c=-b$. From there it follows that $c=0$, so $0$ is the only constant in $\mathrm{Span}(\sin x,\cos x)$.
