Background
In complex plane, x-axis, y-axis are $\mathbb{R}$ and imaginary number $i=\sqrt{-1}$ respectively. The complex number $2+3i$ expressed as a vector form $(0,0)$ to $(2,3)$. Let two unit vector $\overset{\rightharpoonup }{z_1}, \overset{\rightharpoonup }{z_2}$
$$\begin{align*} & \overset{\rightharpoonup }{z_1} = \cos{\theta_1}+ i\sin{\theta_1} = (\cos{\theta_1}, \sin{\theta_1}) \\ & \overset{\rightharpoonup }{z_2} = \cos{\theta_2}+ i\sin{\theta_2} = (\cos{\theta_2}, \sin{\theta_2}) \end{align*}$$Identities in Complex Plane
In complex plane, vector $z_1 \cdot z_2$ with length $|z_1||z_2|$ and angel $\theta_1+ \theta_2$. We write the math equation
$$\begin{align*} \overset{\rightharpoonup }{z_1} \cdot \overset{\rightharpoonup }{z_2} &= \cos{(\theta_1+\theta_2)}+ i\sin{(\theta_1+\theta_2)} \\ &= (\cos{\theta_1}+ i\sin{\theta_1}) \cdot (\cos{\theta_2}+ i\sin{\theta_2}) \\ &= (\cos{\theta_1} \cos{\theta_2} +i^2 \sin{\theta_1} \sin{\theta_2}) + i(\cos{\theta_1} \sin{\theta_2}+ \sin{\theta_1} \cos{\theta_2}) \\ &= (\cos{\theta_1} \cos{\theta_2} - \sin{\theta_1} \sin{\theta_2}) + i(\cos{\theta_1} \sin{\theta_2}+ \sin{\theta_1} \cos{\theta_2}) \end{align*}$$Therefore, we explain the formula
$$\begin{align*} & \cos{(\theta_1+\theta_2)} = \cos{\theta_1} \cos{\theta_2} - \sin{\theta_1} \sin{\theta_2} \\ & \sin{(\theta_1+\theta_2)} = \cos{\theta_1} \sin{\theta_2}+ \sin{\theta_1} \cos{\theta_2} \end{align*}$$