A solution (x,y)of the system of equations x−y=13and cos2(πx)−sin2(πy)=12 given by

x−y=13;cos2πx−sin2πy=12cosπ(x+y)cos(π(13))=12cosπ(x+y)=2nπ=1∴x+y=2x−y=16∴x=2+16,y=2−16(x,y)=136,116

A solution (x,y)of the system of equations x−y=13and cos2(πx)−sin2(πy)=12 given by

x−y=13;cos2πx−sin2πy=12cosπ(x+y)cos(π(13))=12cosπ(x+y)=2nπ=1∴x+y=2x−y=16∴x=2+16,y=2−16(x,y)=136,116

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A solution $(x, y)$ of the system of equations $x - y = \frac{1}{3}$ and $\cos^{2} (π x) - \sin^{2} (π y) = \frac{1}{2}$ given by

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$(\frac{2}{3}, \frac{1}{3})$

$(\frac{7}{6}, \frac{1}{6})$

$(\frac{13}{6}, \frac{11}{6})$

$(\frac{1}{6}, \frac{5}{6})$

answer is C.

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Detailed Solution

$x - y = \frac{1}{3}; \cos^{2} π x - \sin^{2} π y = \frac{1}{2}$

$\cos π (x + y) \cos (π (\frac{1}{3})) = \frac{1}{2}$

$\cos π (x + y) = 2 n π = 1$

$∴ x + y = 2$

$x - y = \frac{1}{6}$

$∴ x = 2 + \frac{1}{6}, y = 2 - \frac{1}{6}$

$(x, y) = (\frac{13}{6}, \frac{11}{6})$

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A solution (x,y)of the system of equations x−y=13and cos2(πx)−sin2(πy)=12 given by

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A solution $(x, y)$ of the system of equations $x - y = \frac{1}{3}$ and $\cos^{2} (π x) - \sin^{2} (π y) = \frac{1}{2}$ given by