show the function $\cos^2(x)$ and $\sin^2(x)$ belong to Trig2(R) and find their coordinates for the basis In the next question it  can be used without proof that the familie 
$(1,\cos(x),\sin(x),\cos(2x),\sin(2x))$ is lineart independent and thus a basis for trig2(R) 
b) show that the function $\cos^2(x)$ and $\sin^2(x)$ belong to Trig2(R) and determine their coordinate for the basis 
I know  $\cos(2x) = \cos^2(x)- \sin^2(x)$ but how about $\sin(2x)= 2\cos(x)\sin(x)$

$ \cos (2x)= \cos^2 x- \sin^2x = \cos^2 x-(1- \cos^2 x)= 2 \cos^2 x-1,$ hence
$$ \cos^2 x= \frac{1}{2}(  \cos(2x)-1).$$
Can you proceed ?
