Find the lines tangent and normal to \( y= 2\sin \left ( \pi x - y \right ) \) at \( (1,0) \)

\( \frac{dy}{dx} = 2 \cos \left ( \pi x - y \right ) \left ( \pi - \frac{dy}{dx} \right ) \)

\( \frac{dy}{dx} = 2\pi \cos \left ( \pi x - y \right ) - 2 \cos \left ( \pi x - y \right ) \frac{dy}{dx} \)

\( \frac{dy}{dx}+2 \cos \left ( \pi x - y \right ) \frac{dy}{dx} = 2\pi \cos \left ( \pi x - y \right ) \)

\( \left ( 1 + 2\cos \left ( \pi x - y \right ) \right ) \frac{dy}{dx} = 2\pi \cos \left ( \pi x - y \right ) \)

\( \frac{dy}{dx} = \frac{ 2\pi \cos \left ( \pi x - y \right ) }{1 + 2\cos \left ( \pi x - y \right ) } \)

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\( \frac{dy}{dx}|_{(x,y)=(1,0)} = \frac{ 2\pi \cos \left ( \pi (1) - 0 \right ) }{1 + 2\cos \left ( \pi (1) - 0 \right ) } = \frac{ 2 \pi (-1)}{1+2(-1)}=2\pi\)

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Tangent Line: \( y-0 = 2\pi (x-1) \)

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Normal Line: \( y-0 = \frac{-1}{2\pi} (x-1) \)

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