sqing wrote: Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$ Squaring $\sin u+\cos u=\pm 1$, we get $\sin u\cos u=0$ and so $u=k\frac{\pi}2$ for some $k\in\mathbb Z$ Setting $\sqrt x=\frac k2$, equation becomes $\sin \frac{k\pi}2+\cos\frac{k\pi}2=(-1)^{\left\lfloor\frac k2\right\rfloor}$ And so we only have to check $ k$ modulo 4, which all fit. Hence the answer : $\boxed{x=\frac{k^2}4\text{ whatever is }k\in\mathbb Z_{\ge 0}}$