Review of a proof for De Moivre's theorem using mathematical induction

Solution 1:

You're very close!

The first issue I see is a minor one. The identity $$k\sin(n \theta+\theta)=k\sin n \theta\cdot\cos \theta + \cos n\theta\cdot k\sin\theta$$ seems to be unnecessary. I just can't tell what use you make of it, since the preceding identity does what you need.

The second issue I see is a major one. The identity $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta+\sin n\theta\cdot\sin\theta$$ is incorrect. It should instead be $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta-\sin n\theta\cdot\sin\theta,$$ which allows you to draw the desired conclusion. (I suspect this may have been a typo.)