Section 8.8 Exercise 19

Use the Taylor series given in Key Idea 8.8.1 to verify the given identity:

\[ \frac{d}{dx}(sin \ x) = cos \ x \]

Recall the series representation of sin x:

\[ sin \ x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \]

\[ \frac{d}{dx}(sin \ x) = \frac{d}{dx} \ (x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...) \]

Differentiate and rewrite factorials as \(n! = n (n-1)!\)

\[ \frac{d}{dx}(sin \ x) = 1 - \frac{3x^2}{3\times2!} + \frac{5x^4}{5\times4!} - \frac{7x^6}{7\times6!} + ... \]

\[ \frac{d}{dx}(sin \ x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... \]

\[ \frac{d}{dx}(sin \ x) = cos\ x \]

The identify is verified.