The value of \(\cos^3(\pi/8)\cdot\cos(3\pi/8) + \sin^3(\pi/8)\cdot\sin(3\pi/8)\) is:
1/(2√2)
Since \(3\pi/8 = \pi/2 - \pi/8\), we have \(\cos(3\pi/8) = \sin(\pi/8)\) and \(\sin(3\pi/8) = \cos(\pi/8)\).
Substituting, the expression becomes \(\cos^3(\pi/8)\sin(\pi/8) + \sin^3(\pi/8)\cos(\pi/8)\).
Factoring out \(\sin(\pi/8)\cos(\pi/8)\): \(= \sin(\pi/8)\cos(\pi/8)\left[\cos^2(\pi/8)+\sin^2(\pi/8)\right] = \sin(\pi/8)\cos(\pi/8)\), since \(\cos^2\theta+\sin^2\theta=1\).
Using the double angle identity, \(\sin(\pi/8)\cos(\pi/8) = \tfrac{1}{2}\sin(\pi/4) = \tfrac{1}{2}\cdot\tfrac{\sqrt2}{2} = \dfrac{1}{2\sqrt2}\).
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