Proving that $\sin^7\theta + \cos^7\theta <1$ using basic trigonometry and identities How do I prove $\sin^7\theta + \cos^7\theta < 1$ for an angle between $(0,\pi/2)$?

Hint: If $\theta \in (0, \pi/2)$, we have $$0 < \sin \theta < 1 \text{ and }0 < \cos \theta < 1,$$ which gives us that $\sin^7 \theta < \sin^2\theta$ and $\cos^7\theta < \cos^2\theta$. So...
