Example 3 Find the sum: (1) $\cos ^{2} 1^{\circ}+\cos ^{2} 3^{\circ}+\cdots+\cos ^{2} 89^{\circ}$
(2) Let $f(x)=\frac{4^{x}}{4^{x}+2}$, find $f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)$
Analysis and Solution The key to solving this problem lies in recognizing that these two sequences satisfy $\cos ^{2} k^{\circ}+\cos ^{2}\left(90^{\circ}-k^{\circ}\right)=1$, $f(x)+f(1-x)=1$.  
Using the reverse order addition method, the sum of (1) is $\frac{89}{2}$, and the sum of (2) is 500.