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.

I do not think there is a closed form for all values of $k$, but one can play around with trigonometric identities to simplify the expression for certain values of $k$. For instance:


*

*If $k=2$, then:


$$\sin^4 x + \cos^4 x = (1-\cos^2 x)^2 + \cos^4 x\\
= 1-2\cos^2x + 2\cos^4 x \\
= 1-2\cos^2x(1-\cos^2x)\\
= 1-2\sin^2x\cos^2x\\
= 1 - \frac{\sin^2(2x)}{2}.$$


*

*If $k=3$, then:


$$\sin^6 x + \cos^6 x = (1-\cos^2 x)^3 + \cos^6 x\\
= 1-3\cos^2x + 3\cos^4 x - \cos^6 x + \cos^6 x \\
= 1-3\cos^2x + 3\cos^4x\\
= 1-3\cos^2x(1-\cos^2x)\\
= 1-3\sin^2x\cos^2x\\
= 1 - \frac{3\sin^2(2x)}{4}.$$
