I want to show that aj in this photo, but I am unsure of how to get there I know it is a minimum so it takes differentiation to get there but I am unsure how, any help would be appreciated, thanks enter image description here
$$f(x)=\frac{a_0}{2}T_0(x)+\sum\limits_{j=1}^{\infty } a_j\ T_j(x)\tag{1}$$
$$a_j=\frac{2}{\pi}\int_{-1}^1 f(x)\ T_j(x)\frac{dx}{\sqrt{1-x^2}}\quad (j=0,1,2,...)\tag{2}$$