> [!NOTE] > For all $n\geq 1, x_{1},x_{2},\dots,x_{n}\in \mathbb{R},$ $|\sin x_{1}| + |\sin x_{2}| + \dots + |\sin x_{n}| + | \cos (x_{1}+x_{2}+\dots+x_{n})| \geq 1.\tag{1}$ ###### Proof For all $x\in \mathbb{R},$ $( |\sin x| + |\cos x|)^2 = \sin^2 x + \cos^2 x +|2\sin x \cos x| = 1 +|\sin 2x| \geq 1$thus $|\sin x|+ |\cos x|\geq 1$that is, $(1)$ holds true for $n=1.$ Suppose $(1)$ holds true for $n=k.$ STS $|\sin x_{k+1}|+|\cos(x_{1}+x_{2}+\dots.+x_{k+1})| \geq |\cos (x_{1}+x_{2}+\dots+x_{k})| \tag{2}$since this implies $(1)$ holds true for $n=k+1$ (add $|\sin x_{1}| + |\sin x_{2}| + \dots + |\sin x_{n}|$ to both sides) and then by [[Induction Principle|mathematical induction]], $(1)$ holds true for all $n$ as above. Let $x=x_{1}+\dots+x_{n+1}$ and $y=x_{n+1}.$ Then by [[Cosine of Sum]], $\begin{align} |\cos(x-y)| &= |\cos x \cos y + \sin x \sin y| \\ & \leq |\cos x| | \cos y| + |\sin x| |\sin y| \\ & \leq |\cos x| + |\sin y| \end{align}$which is exactly $(2).$ ###### Proof As shown above, for all $x\in \mathbb{R},$ $|\sin x|+ | \cos x| \geq 1 \tag{3}$ Let $n \geq 1, x_{1},x_{2},\dots,x_{n}\in \mathbb{R}.$ Since [[Absolute Value of Sine is Subadditive]], $| \sin (x_{1}+x_{2}+\dots+x_{n})| \leq |\sin x_{1}| + | \sin x_{2}| +\dots+ | \sin x_{n}|$ Then $\begin{align} &|\sin x_{1}| + |\sin x_{2}| + \dots + |\sin x_{n}| + | \cos (x_{1}+x_{2}+\dots+x_{n})| \\ &\geq |\sin (x_{1}+x_{2}+\dots+x_{n})| +| \cos (x_{1}+x_{2}+\dots+x_{n})| \\ &\geq 1 \end{align}$using $(3)$ with $x= x_{1}+x_{2}+\dots+x_{n}.$