checking if a function lies in the span of a vector

Question

the span I'm given is

$$\text{span}(\cos^2x,\sin^2x)$$

I'm working in $F(0,\pi)$

I need to check if $\cos2x$ and $x^2$ lie in that span. How would I go about doing this.

Answer 1

The double angle formula for $\cos 2x$ will come in handy to show that $\cos 2x$ is in the span. Next suppose that towards a contradiction there exists $a,b\in\mathbb{R}$ such that for all $x\in[0,\pi]$ $$ a\cos^2x+b\sin^2x=x^2\tag{1}. $$ Set $x=0$, to find $a$, and then set $x=\pi/2$ to find $b$. Now substitute $x=\pi/6$, in (1). Are both sides equal?