20. The inequality to be proved is
$$\sin ^{2 n} x+\left(2^{n}-2\right) \sin ^{n} x \cos ^{n} x+\cos ^{2 n} x \leqslant 1$$

Taking the $n$-th power on both sides of the identity $\sin ^{2} x+\cos ^{2} x=1$, we get
$$\begin{aligned} 1= & \left(\sin ^{2 n} x+\cos ^{2 n} x\right)+n\left(\sin ^{2} x \cos ^{n-2} x+\cos ^{2} x \sin ^{n-2} x\right)+ \\ & \mathrm{C}_{n}^{2}\left(\sin ^{4} x \cos ^{n-4} x+\cos ^{4} x \sin ^{n-4} x\right)+\cdots \geqslant \\ & \left(\sin ^{2 n} x+\cos ^{2 n} x\right)+\left(2^{n}-2\right) \sin ^{n} x \cos ^{n} x \end{aligned}$$

This is because the value in each parenthesis is no less than $2 \sin ^{n} x \cos ^{n} x$, and the sum of the coefficients equals $\frac{1}{2}\left(2^{n}-\right.$