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How to Compute $\frac{\frac{1}{4} \sin(2\theta)}{\frac{1}{4} \sin(\theta+\theta)}$?
I am having trouble trying to compute this line.
$$\frac{\frac{1}{4} \sin(2\theta)}{\frac{1}{4} \sin(\theta+\theta)}$$
I tried using the complex half-angle formula but I couldn’t solve it.
A:
This problem can be solved by completing the $2\theta$ portion of the sine sign and using the double angle formula. Let $$\sin z = \left(\frac{\cos z + i\sin z}{1 – i\cos z}\right)^2$$
Thus, $$\frac{\sin^2 2\theta}{\sin\left(\theta+2\theta\right)}=\left(\frac{\frac{1}{4}\sin 2\theta}{\frac{1}{4}\sin\left(\theta+2\theta\right)}\right)^2=\left(\frac{\cos^2 2\theta + \sin^2 2\theta}{\cos\left(\theta+2\theta\right) + \sin\left(\theta+2\theta\right)}\right)^2=$$ $$\frac{1+\cos 4\theta + 2\cos 2\theta\sin 2\theta}{1+\cos 2\theta – \cos 4\theta + 2\cos 2\theta\sin 2\theta}=\frac{1+\cos 4\theta + 2\cos 2\theta\sin 2\theta}{1 + \cos 4\theta – \cos 2\theta + \cos 2\theta – \cos 2\theta – \cos 2\theta + \cos 2\theta}=$$ $$\frac{1+\cos 4\theta
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