$\displaystyle (1 + cos{\not2\theta} + isin\theta )^n + (1+cos\theta - isin\theta )^n = 2^{n+1}cos^n\frac{\theta }{2}cos\frac{n\theta }{2 } $ (The 2 on the left should not be there.)
Use the identities $\displaystyle 1+\cos\theta=2\cos^2\tfrac\theta2$ and $\displaystyle \sin\theta = 2\sin\tfrac\theta2\cos\tfrac\theta2$.