From power mean inequality $\rightarrow~~\sqrt[n]{\frac{(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n}{2}} \geq \frac{1+\sin^2\alpha+1+\cos^2\alpha}{2}$ $=\frac{1+1+(\sin^2\alpha+\cos^2\alpha)}{2}=\frac{3}{2}$ There for $\frac{(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n}{2} \geq (\frac{3}{2})^n$ $\therefore~~(1+\sin^2\alpha)^n+(1+\cos^2\alpha)^n\geq 2(\frac{3}{2})^n~~\forall n \in \mathbb{N} $ Equality case holds at $sin^2\alpha=cos^2\alpha=\frac{1}{2}$ or $n=1$