Prove by method of finite induction for n belongs to Natural Numbers

Question

I cannot prove the third step; I don't understand how to go ahead!

Answer 1

Are you familiar with the angle sum formulas for $\sin$ and $\cos$? For reference, $$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$$ $$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$$ You should be able to apply these rules to the right hand side of the statement in part c) in order to write it in terms of your base case and assumption in part b).