In Exercises 17 – 20, use the Taylor series given in Key Idea 32 to verify the given identity.
The series for \(sin(x)\) is: \[sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\]
Starting with \(sin(-x)\) we obtain: \[sin(-x)=\sum_{n=0}^{\infty} (-1)^n \frac{(-x)^{2n+1}}{(2n+1)!}\] \[= \sum_{n=0}^{\infty} (-1)^{n+1} \frac{x^{2n+1}}{(2n+1)!}\]
\[= -\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = -sin(x)\]