Proof of RagvaloD: $|\cos x|+|\cos 2x| \geq \frac{\sqrt{2}}{2}$ If $n=2k-1$ then $|\cos x|+...+|\cos 2^n x|\geq k\frac{\sqrt 2}{2}=\frac{n+1}{2\sqrt{2}}>\frac{n}{2\sqrt{2}}$ If $n=2k$ then $ |\cos x|+...+|\cos 2^n x|\geq k\frac{\sqrt 2}{2}+|\cos x| \geq \frac{n}{2\sqrt{2}}$

For every real number $ x$ prove the inequality: $$|(\sin x-1)(\cos x+1)-\sin^2x|\leq \frac{9+3\sqrt 3}{4}$$