The cosine function 𝑐𝑜𝑠(𝑥) can be expanded into a Taylor series:

• $cos(x)=\frac{x^0}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}...$

This is also a power series expansion of the cosine function

Proof

We start with the definition of a Taylor series

• $f(x)=\frac{f(a)(x-a{)}^0}{0!}+\frac{f^{\prime }(a)(x-a{)}^1}{1!}+\frac{f^{\prime \prime }(a)(x-a{)}^2}{2!}+\frac{f^{\prime \prime \prime }(a)(x-a{)}^3}{3!}+\frac{f^{\prime \prime \prime \prime }(a)(x-a{)}^4}{4!}+...$
• $f(x)=\frac{f(0)x^0}{0!}+\frac{f^{\prime }(0)x^1}{1!}+\frac{f^{\prime \prime }(0)x^2}{2!}+\frac{f^{\prime \prime \prime }(0)x^3}{3!}+\frac{f^{\prime \prime \prime \prime }(0)x^4}{4!}+...\text{ let }a\text{ = 0}$
• $cos(x)=\frac{cos(0)x^0}{0!}+\frac{co{s}^{\prime }(0)x^1}{1!}+\frac{co{s}^{\prime \prime }(0)x^2}{2!}+\frac{co{s}^{\prime \prime \prime }(0)x^3}{3!}+\frac{co{s}^{\prime \prime \prime \prime }(0)x^4}{4!}+...\text{ let }f(x)=cos(x)$
• $cos(x)=\frac{cos(0)x^0}{0!}+\frac{-sin(0)x^1}{1!}+\frac{-cos(0)x^2}{2!}+\frac{sin(0)x^3}{3!}+\frac{cos(0)x^4}{4!}+...\text{ evaluate derivatives}$
• $cos(x)=\frac{1x^0}{0!}+\frac{0x^1}{1!}+\frac{-1x^2}{2!}+\frac{0x^3}{3!}+\frac{1x^4}{4!}+...$
• $cos(x)=1+0-\frac{x^2}{2!}+0+\frac{x^4}{4!}+...$
• $cos(x)=1+0-\frac{x^2}{2}+0+\frac{x^4}{24}+...$
• $cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+...$