Prove that $\left(a\right){\left(1+i\right)}^n+{\left(1-i\right)}^n=2^{\frac{n+2}{2}}.\cos \left(\frac{n\pi }{4}\right)$, where n is a positive integer. $\left(b\right){\left(1+i\sqrt{3}\right)}^n+(1-i{\sqrt{3}}^n=2^{n+1}\cos \left(\frac{n\pi }{3}\right)$, where n is a positive integer

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