Taylor series of trigonometric functions

From Maclaurin series of $\sin(x)$ :

$\large \displaystyle \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Dividing by $x$ on both sides:

$\large \displaystyle \frac{\sin(x)}{x} = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n+1)!}$

Differentiating with respect to $x$ on both sides:

$\large \displaystyle \frac{x\cos(x) - \sin(x)}{x^2} = 2\sum_{n=0}^{\infty} \frac{(-1)^n n x^{2n-1}}{(2n+1)!}$

Multiplying by $x^2$ on both sides:

$\large \displaystyle x\cos(x) - \sin(x) = 2\sum_{n=0}^{\infty} \frac{(-1)^n n x^{2n+1}}{(2n+1)!}$

$\large \displaystyle x\cos(x) - \sin(x) = 2 \left [ \sum_{n=0}^{\infty} \frac{(-1)^n n x^{2n+1}}{(2n+1)!} \color{#20A900} + \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} - \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \right ]$

$\large \displaystyle x\cos(x) - \sin(x) = 2 \left [ \sum_{n=0}^{\infty} \frac{(-1)^n (n+1) x^{2n+1}}{(2n+1)!} - \sin(x) \right ]$

$\large \displaystyle x\cos(x) + \sin(x) = 2 \ \sum_{n=0}^{\infty} \frac{(-1)^n (n+1) x^{2n+1}}{(2n+1)!}$

$\color{#20A900} \boxed{\large \displaystyle S(x) = x\cos(x) + \sin(x) }$

So:

$\color{#3D99F6} \boxed{\large \displaystyle S(1) = \cos(1) + \sin(1) \approx 1.38 }$

$\displaystyle \begin{aligned} S(x) & = {\color{#20A900}{2}}\sum_{n \ = \ 0}^{\infty} \frac{(- 1)^n(n + 1)x^{2n + 1}}{(2n + 1)!} & \small \color{#3D99F6} c\sum a_n = \sum c \cdot a_n \ \text{for convergent series} \\ & = \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^n({\color{#20A900}{2n + 2}})x^{2n + 1}}{(2n + 1)!} \\ & = \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^n\Bigg[\bigg((2n + 1) + 1\bigg)x^{2n + 1}\Bigg]}{(2n + 1)!} & \small \color{#3D99F6} \text{Distribute} \\ & = \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^n({\color{#D61F06}{2n + 1}})x^{2n + 1}}{({\color{#D61F06}{2n + 1}})!} + \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^nx^{2n + 1}}{(2n + 1)!} & \small \color{#3D99F6} \frac{2n + 1}{(2n + 1)!} = \frac{1}{(2n)!} \\ & = \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^nx^{2n + 1}}{(2n)!} + \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^nx^{2n + 1}}{(2n + 1)!} & \small \color{#3D99F6} x^{2n + 1} = x \cdot x^{2n} \\ & = \sum_{n \ = \ 0}^{\infty} x\frac{(- 1)^nx^{2n}}{(2n)!} + \sum_{n \ = \ 0}^{\infty} \frac{(- 1)^nx^{2n + 1}}{(2n + 1)!} \\ & = x\cos x + \sin x \\ S(1) & = \cos (1) + \sin (1) \end{aligned}$