Answer
$X=x \cos \phi+y\sin \phi$ and $Y=-x \sin \phi+y\cos \phi$
Work Step by Step
Need to multiply the first equation by $\cos \phi$ and the second equation by $\sin \phi$.
$x \cos \phi=X\cos^2 \phi-Y \sin \phi \cos \phi$
and $y \sin \phi=X\cos^2 \phi+Y \sin \phi \cos \phi$
After adding the both equations we get $X=x \cos \phi+y\sin \phi$
Further, need to multiply the first given equation by $-\sin \phi$ and the second given equation by $\cos \phi$.
$-x \sin \phi=-X \sin \phi \cos \phi+Y\sin^2 \phi$
and $y \cos \phi=X \sin \phi \cos \phi+Y\cos^2 \phi$
After adding the both equations we get
$Y=-x \sin \phi+y\cos \phi$
Hence, $X=x \cos \phi+y\sin \phi$ and $Y=-x \sin \phi+y\cos \phi$
