What does  $\sin^{2k}\theta+\cos^{2k}\theta=$? 
What is the sum $\sin^{2k}\theta+\cos^{2k}\theta$ equal to? 

Besides Mathematical Induction,more solutions are desired.

If you let $z_k=\cos^k(\theta)+i\sin^k(\theta)\in\Bbb C$, it is clear that
$$
\cos^{2k}(\theta)+\sin^{2k}(\theta)=||z_k||^2.
$$
When $k=1$ the complex point $z_1$ describes (under the usual Argand-Gauss identification $\Bbb C=\Bbb R^2$) the circumference of radius $1$ centered in the origin, and your expression gives $1$.
For any other value $k>1$, the point $z_k$ describes a closed curve $\cal C_k\subset\Bbb R^2$ and your expression simply computes the square distance of the generic point from the origin. There's no reason to expect that this expression may take a simpler form than it already has. 
