Step-by-step explanation:
Ruler's formula states that
[tex]e^{i\theta} = \cos{\theta} +i\sin{\theta}[/tex]
We also know that
[tex]e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1+\theta_2)}[/tex]
therefore,
[tex]e^{i(\theta_1+\theta_2)}= \cos{(\theta_1+\theta_2)}+ \sin{(\theta_1+\theta_2)}[/tex] (1)
Similarly, we can write
[tex]e^{-i(\theta_1+\theta_2)} = \cos{(\theta_1+\theta_2)} - \sin{(\theta_1+\theta_2)}[/tex] (2)
Adding Eqn(1) and Eqn(2) together, we get
[tex]2\cos{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}[/tex]
or
[tex]\cos{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} + e^{-i(\theta_1+\theta_2)}}{2}[/tex]
To get the expression for the sine function, we subtract Eqn(2) from Eqn(1) to get
[tex]2i\sin{(\theta_1+\theta_2)} = e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}[/tex]
or
[tex]\sin{(\theta_1+\theta_2)} = \dfrac{e^{i(\theta_1+\theta_2)} - e^{-i(\theta_1+\theta_2)}}{2i}[/tex]
