Use the polar form of a complex number to show that for any two numbers α and β

Question

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Okay so I have a complex number question here I'm very stumped with. It is as follows:

Use the polar form of a complex number to show that for any two
numbers α and β the following identities hold:

sin(α + β) = sin(α) cos(β) + cos(α) sin(β)  
cos(α + β) = cos(α) cos(β) − sin(α) sin(β)

How would I go about doing this?

Answer 1

Let $$ w=e^{i\alpha}=\cos\alpha+i\sin\alpha\\ z=e^{i\beta}=\cos\beta+i\sin\beta $$

Multiplying: $$\begin{align} wz=e^{i\alpha}\cdot e^{i\beta}&=(\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta)\\ e^{i(\alpha+\beta)}&=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\\ \text{Using De Moivre's theorem,}\hspace{2cm}\\ \cos(\alpha+\beta)+i\sin(\alpha+\beta)&=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\\ \text{Equating real and imaginary parts}\hspace{1cm}\\ \cos(\alpha+\beta)&=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\ \sin(\alpha+\beta)&=\sin\alpha\cos\beta+\cos\alpha\sin\beta \end{align}$$

Answer 2

HINT: Try representing $z_{1} = e^{i\alpha}$ and $z_{2} = e^{i\beta}$, and remember Euler's formula: $e^{i\theta} = \cos(\theta) + i\sin(\theta)$.

Then remember that the real and imaginary parts are independent.