# Given that it appears impossible to make the set out of linear combinations of its elements, why is it still dependent?

The answer key says the following set of functions is linearly dependent: $\{5, \cos^2x, \sin^2x\}$.

Without calculating the Wronskian, I would've guessed it was independent because there's apparently no way you can form a linear combination out of any of these functions to get others: you can't multiply $5$ to get $\cos x$; you can't multiply $\cos x$ to get $\sin x$, etc. What's wrong with my reasoning?

• It is not $\cos x$ and $\sin x$, but $(\cos x)^2$ and $(\sin x)^2$. – Marc van Leeuwen Mar 15 '15 at 17:28

You can use $\sin^2x+\cos^2x=1$ to form a linear combination of those three functions that results in $0$.
you can't multiply 5 to get $\cos x$, you can't multiply $cos$ to get $\sin x$ etc. What's wrong with my reasoning?
But linear combinations are, in general, combinations — you need to consider sums of scalar multiples of the functions. And (as other answers show) there is a way to get 5 as a sum of scalar multiples of $\cos^2 x$ and $\sin^2 x$: $$5 = 5 \cos^2 x + 5 \sin^2 x$$ or, rearranged into the symmetric form of a linear dependency: $$(-1) \cdot 5 + 5 \cdot \cos^2 x + 5 \cdot \sin^2 x = 0.$$
Hint: $$5 \cos^2 x+ 5\sin^2x=5$$
You have $-\frac{1}{5}*5+\cos^2(x)+\sin^2(x)=-1+1=0$.