How can I solve this problem?

Question

A rotation φ1 + φ2 about the z-axis is carried out as two successive rotations φ1 and φ2, each about the z-axis. Use the matrix representation of the rotations to derive the trigonometric identities

Answer 1

$Mat(\phi_1)=\begin{pmatrix} \cos(\theta_1) & -\sin(\theta_1) \\ \sin(\theta_1) & \cos(\theta_1) \end{pmatrix}$ and $Mat(\phi_2)=\begin{pmatrix} \cos(\theta_2) & -\sin(\theta_2) \\ \sin(\theta_2) & \cos(\theta_2) \end{pmatrix}$

$$Mat(\phi_1 \phi_2) = \begin{pmatrix} \cos(\theta_1)\cos(\theta_2)-\sin(\theta_1)\sin(\theta_2) & -\sin(\theta_2)\cos(\theta_1)-\sin(\theta_1)\cos(\theta_2) \\ \sin(\theta_1)\cos(\theta_2)+\cos(\theta_1)\sin(\theta_2) & -\sin(\theta_1)\sin(\theta_2)+\cos(\theta_1)\cos(\theta_2) \end{pmatrix}$$ It is the matrix of the composition of the 2 rotations around the same axis,so it is the matrix around axis z of angle $\theta_1+\theta_2$

So you have, by identification : $$\cos(\theta_1+\theta_2)=\cos(\theta_1)\cos(\theta_2)-\sin(\theta_1)\sin(\theta_2)$$ $$\sin(\theta_1+\theta_2)=\sin(\theta_2)\cos(\theta_1)+\sin(\theta_1)\cos(\theta_2)$$