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Period of \sin(x) is 2\pi
Period of \cos(x) is 2\pi

Adding two functions having 2\pi as period produces another function having the same 2\pi as a period, so the period of \sin(x)+\cos(x) is also 2\pi.

|\sin(x)+\cos(x)| flips the negative part over x-axis giving a shorter period : \boxed{\pi}
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