Find the Maclaurin series for f(x)=cos2(x)f(x) = \cos^2(x)f(x)=cos2(x) using the identity cos2(x)=1+cos(2x)2\cos^2(x) = \frac{1+\cos(2x)}{2}cos2(x)=21+cos(2x).
1+∑n=1∞(−1)n22n−1x2n(2n)!1 + \sum_{n=1}^{\infty} \frac{(-1)^n 2^{2n-1} x^{2n}}{(2n)!}1+∑n=1∞(2n)!(−1)n22n−1x2n
∑n=0∞(−1)n22nx2n(2n)!\sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n} x^{2n}}{(2n)!}∑n=0∞(2n)!(−1)n22nx2n
1+∑n=1∞(−1)n22nx2n(2n)!1 + \sum_{n=1}^{\infty} \frac{(-1)^n 2^{2n} x^{2n}}{(2n)!}1+∑n=1∞(2n)!(−1)n22nx2n
∑n=1∞(−1)n22n−1x2n(2n)!\sum_{n=1}^{\infty} \frac{(-1)^n 2^{2n-1} x^{2n}}{(2n)!}∑n=1∞(2n)!(−1)n22n−1x2n