Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at $(-2,0)$ and $(2,0)$; vertices at $(1,0)$ and $(-1,0)$

Step 1 ：The standard form of the equation of a hyperbola with center at the origin is given by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) if the hyperbola opens horizontally, and \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) if it opens vertically.

Step 2 ：The distance from the center to a focus is \(c\), and the distance from the center to a vertex is \(a\). We also know that \(c^2 = a^2 + b^2\).

Step 3 ：Given that the foci are at \((-2,0)\) and \((2,0)\), we can find that \(c = 2\).

Step 4 ：Given that the vertices are at \((-1,0)\) and \((1,0)\), we can find that \(a = 1\).

Step 5 ：We can substitute \(a\) and \(c\) into the equation \(c^2 = a^2 + b^2\) to find \(b\).

Step 6 ：Now that we have the values of \(a\), \(b\), and \(c\), we can substitute them into the standard form of the equation of a hyperbola. Since the foci and vertices are on the x-axis, the hyperbola opens horizontally, so the equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 7 ：The standard form of the equation of the hyperbola is \(\boxed{x^2 - \frac{y^2}{3} = 1}\).