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.

use Holder inequality
$$[(\cos^2{x})^{n+1}+(\sin^2{x})^{n+1}][1+1]^{n} \geq (\cos^2{x}+\sin^2{x})^{n+1}=1$$
or
Use AM-GM inequality we have
$$(\sin^2{x})^{n+1}+\dfrac{1}{2^{n+1}}+\dfrac{1}{2^{n+1}}+\cdots+\dfrac{1}{2^{n+1}}\ge (n+1)\cdot\sqrt[n+1]{\dfrac{1}{2^{n(n+1)}}(\sin^2{x})^{n+1}}=\dfrac{n+1}{2^{n}}\sin^2{x}$$
so
$$(\sin^2{x})^{n+1}+\dfrac{n}{2^{n+1}}\ge\dfrac{n+1}{2^{n+1}}\sin^2{x}\tag{1}$$
the same as
$$(\cos^2{x})^{n+1}+\dfrac{n}{2^{n+1}}\ge\dfrac{n+1}{2^{n+1}}\cos^2{x}\tag{2}$$
$(1)+(2)$
$$\sin^{2n+2}{x}+\cos^{2n+2}{x}\ge\dfrac{1}{2^n}$$
