证明1+1=2

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\begin{aligned}1+1&=(\sin\frac{\pi}{2}-\sin0)+(\cos0-\cos\frac{\pi}{2})\\&=\sin x\huge\mid_{0}^{\frac{\pi}{2}}\normalsize+\cos x\huge\mid_{\frac{\pi}{2}}^{0}\normalsize\\&=\displaystyle \int_0^{\frac{\pi}{2}}\cos xdx+\displaystyle \int_{\frac{\pi}{2}}^0(-\sin x)dx\\&=\displaystyle \int_0^{\frac{\pi}{2}}(\cos x+\sin x)dx\\&=\displaystyle \int_0^{\frac{\pi}{2}}\sqrt2\sin(x+\frac{\pi}{4})dx\\&=-\sqrt2\cos(x+\frac{\pi}{4})\huge\mid_0^{\frac{\pi}{2}}\\&=-\sqrt2(\cos(\frac{\pi}{2}+\frac{\pi}{4})-\cos\frac{\pi}{4})\\&=\sqrt2(\sin\frac{\pi}{4}+\cos\frac{\pi}{4})\\&=\sqrt2\times\sqrt2\sin(\frac{\pi}{4}+\frac{\pi}{4})\\&=2\sin(\frac{\pi}{4}(1+1))\end{aligned} 1+1=2\sin(\frac{\pi}{4}(1+1)) \text{令}x=1+1. x=2\sin(\frac{\pi}{4}x) \text{图像法解方程}

x=-2,0,2 1+1=-2,0,2 1+1\ge2\sqrt{1\times1}=2 1+1=2