To simplify the expression \((2^{-6})^2\), we can use the power of a power property in exponents, which states that \((a^m)^n = a^{m \cdot n}\).
Applying this property, we have:
\[ (2^{-6})^2 = 2^{-6 \cdot 2} = 2^{-12} \]
Now let's look at the multiple-choice answers given:
- \(2^{-42}\)
- \(2^{-12}\)
- \(2^{-6}\)
- \(2^{12}\)
The expression \(2^{-12}\) matches with option 2.
Thus, the equivalent expression for \((2^{-6})^2\) is:
\[ \boxed{2^{-12}} \]