[Math] Prove that $\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$

Question

Prove that $$\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$$

I tried to rationalize the denominator but I always end up with a large fraction that won't cancel out. Is there something I'm missing?

Thanks in advance

Answer 1

This is just a matter of observing that\begin{align}(1+\sin\theta-i\cos\theta)(\sin\theta+i\cos\theta)&=\sin\theta+\sin^2\theta+\cos^2\theta+i(\cos\theta+\sin\theta\cos\theta-\cos\theta\sin\theta)\\&=1+\sin\theta+i\cos\theta.\end{align}