Proving a trigonometric inequality I've been having difficulty with the following,
Prove that,
$[\sin^{n+1}(x)]^{2}+[\cos^{n+1}(x)]^{2} \geq (\frac12)^{n}$ where $x$ is real and $n$ is a non-negative integer.
I've tried an inductive approach but have struggled with the inductive step.

We have $\displaystyle |\sin(x)|\geq \frac{1}{\sqrt {2 }}$ or $\displaystyle |\cos(x)|\geq \frac{1}{\sqrt {2 }}$. Suppose wlog $\displaystyle |\sin(x)|\geq \frac{1}{\sqrt {2 }}$, and put $\displaystyle u=(\sin(x))^2\in [\frac{1}{2},1]=I$. We have then to show that $f(u)=u^{n+1}+(1-u)^{n+1}\geq \frac{1}{2^n}$. The computation of the derivative of $f$ show that on $I$, $f(u)\geq f(1/2)=1/2^n$ and we are done.  
