<s>[INST] Evaluating $ 4\sum^{30}_{n=1} n\;T(n)$, where $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$ 
For $n$ measured in degrees, let 
  $$T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$$
  Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n)$$

I have tried to use double-angle identities but got stuck with the coefficient $n$. I am new to trig, so I probably miss some advanced concepts. 
 [/INST] Like Ginger bread,
Using Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$
$$\cos^2x+\cos^2y=1+\cos(x-y)\cos(x+y)$$
$$\cos x\cos y=\dfrac{\cos(x-y)+\cos(x+y)}2$$
If $\cos(x+y)=\dfrac12\iff x+y=360^\circ m\pm60^\circ,$
$$T(n)=1-\dfrac{\dfrac12}2=?$$
Here $x=30^\circ-n,y=30^\circ+n$
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