Question

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

• A. $(\frac{2}{3},\frac{1}{3})$
• B. $(\frac{7}{6},\frac{1}{6})$
• C. $(\frac{13}{6},\frac{11}{6})$
• D. $(\frac{1}{6},\frac{5}{6})$

Answer 1

The correct option is C $(\frac{13}{6},\frac{11}{6})$

Given $x-y=\frac{1}{3}$ ....(1)
and ${\cos }^2\left(\pi x\right)-{\sin }^2\left(\pi y\right)=\frac{1}{2}$ ...(2)
$\Rightarrow \cos \pi (x+y)\cos \pi (x-y)=\frac{1}{2}$
$\Rightarrow cos\pi (x+y)=1$ (by (1))
$\Rightarrow x+y=2n,n\in I$.
Only option C satisfies above equation