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Cos x ≥ Sin x  during  0 ≤ x ≤ π/4
Cos x ≤ Sin x  for    π/4  ≤ x ≤ π/2

Area shaded :  is split into two regions: one under the Sin x curve and the other under the Cos x curve.

\int\limits^{\frac{\pi}{4}}_0 {Sin\ x} \, dx + \int\limits^{\frac{\pi}{2}}_{\frac{\pi}{4}} {Cos\ x} \, dx \\\\=[-Cos\ x]_0^\frac{\pi}{4}+[Sin\ x]_{\frac{\pi}{4}}^{\frac{\pi}{2}}\\\\=-\frac{1}{\sqrt2}+1+1-\frac{1}{\sqrt2}=2-\sqrt2
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