Independence of tigonometric functions

Question

Are contineous functions

$y=\cos^2 \ (x) ; y = cos(2x) ; y = 1$

linearly dependent or not?

I think that they are, but I cannot explain it in formula.

Generally when we are solving such questions in class, we have same instruction, but we take vectors, such as:

a = (3;1;2) b = (9;-4;2) c = (-1;2;2)

Then it is easy to look for dependance because we can just create a system.

$a= w_1*b+w_2*c$

Now we can just solve the system,

\begin{cases} 9w_1 -w_2 = 3\\ -4w_1 +2w_2 = 1 \\ 2w_1 +2w_2 = 2 \end{cases}

And we find out that $w_1 = 1/2; w_2 = 3/2$

So there is dependance. How do I do the same thing with trigonometric functions?

Answer 1

we know that

$cos(2x)=2cos^2(x)-1$

or $cos(2x)-2cos^2(x)+1=0$.

so the three functions are linearly dependent since there exist $(a,b,c)=(1,-2,1)\neq(0,0,0)$ such that

$a\times cos(2x)+b\times cos^2(x)+c\times 1=0$