The geometric-arithmetic inequality gives
$$\cos x \sin x \le \frac{\cos^2 x + \sin^2 x}{2} = \frac{1}{2}.$$
It follows that
$$1 = (\cos^2 x + \sin^2 x)^2 = \cos^4 x + \sin^4 x + 2 \cos^2 x \sin^2 x \le \cos^4 x + \sin^4 x + \frac{1}{2}$$
so
$$\cos^4 x + \sin^4 x \ge \frac{1}{2} \ge \cos x \sin x.$$
The required inequality follows.