(1) Among the following functions, the one that is an even function with the smallest positive period of $\frac{\pi}{2}$ is ( ).
(A) $y=\sin 2 x+\cos 2 x$
(B) $y=\sin 2 x \cos 2 x$
(C) $y=\sin ^{2} x+\cos 2 x$
(D) $y=\sin ^{2} 2 x-\cos ^{2} 2 x$
(1) D Hint: In option A, taking $x=\frac{\pi}{4}$, then $y=1$; taking $x=-\frac{\pi}{4}$, then $y=-1$, thus $y$ is not an even function; In option B, $y=\frac{1}{2} \sin 4 x$, it is not an even function; In option C, $y=\frac{1+\cos 2 x}{2}$, its smallest positive period is $\pi$; In option D, $y=-\cos 4 x$, it meets the conditions.