\[\begin{aligned} \mathrm{cos}(z+w) &= \mathrm{cos}(z)\mathrm{cos}(w) - \mathrm{sin}(z)\mathrm{sin}(w)\\ \mathrm{sin}(z+w) &= \mathrm{cos}(z)\mathrm{sin}(w) + \mathrm{sin}(z)\mathrm{cos}(w) \end{aligned} \]

Sum Formula for Sin an Cos via Complex Analysis

We have \[ e^i(z+w) = \mathrm{cos}(z+w) + i\mathrm{sin}(z+w) \] and \[ \begin{aligned} e^i(z+w) &= e^{iz} \cdot e^{iw}\\ &= (\mathrm{cos}(z) + i\mathrm{sin}(z))(\mathrm{cos}(w) + i\mathrm{sin}(w))\\ &= \left[ \mathrm{cos}(z)\mathrm{cos}(w) - \mathrm{sin}(z)\mathrm{sin}(w) \right] + i\left[ \mathrm{cos}(z)\mathrm{sin}(w) + \mathrm{sin}(z)\mathrm{cos}(w) \right] \end{aligned} \] compare real and imaginary parts we get \[ \begin{aligned} \mathrm{cos}(z+w) &= \mathrm{cos}(z)\mathrm{cos}(w) - \mathrm{sin}(z)\mathrm{sin}(w)\\ \mathrm{sin}(z+w) &= \mathrm{cos}(z)\mathrm{sin}(w) + \mathrm{sin}(z)\mathrm{cos}(w) \end{aligned} \] as required.