\hspace{-16}$Calculate Sum of \\\\ $\mathbf{(1)\;\; \cos^{2n}1^{\arc0}+\cos^{2n}2^{\arc0}+\cos^{2n}3^{\arc0}+\cos^{2n}4^{\arc0}+..........+\cos^{2n}89^{\arc0}=}$\\\\ $\mathbf{(2)}$ \;\; Prove that $\mathbf{\cos^{10}1^{\arc0}+\cos^{10}2^{\arc0}+\cos^{10}3^{\arc0}+\cos^{10}4^{\arc0}+..........+\cos^{10}89^{\arc0}=\frac{2771}{128}}$
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