To evaluate \(\cos\left(\frac{7\pi}{2}\right)\) using a double-angle formula, it's best to first simplify the angle. The angle \(\frac{7\pi}{2}\) can be simplified by subtracting \(2\pi\) (which is equivalent to \( \frac{4\pi}{2}\)):
\[ \frac{7\pi}{2} - 2\pi = \frac{7\pi}{2} - \frac{4\pi}{2} = \frac{3\pi}{2} \]
Now, we can evaluate \(\cos\left(\frac{3\pi}{2}\right)\). The angle \(\frac{3\pi}{2}\) corresponds to \(270^\circ\), which is located on the negative \(y\)-axis. The cosine of an angle at this position is:
\[ \cos\left(\frac{3\pi}{2}\right) = 0 \]
Therefore,
\[ \cos\left(\frac{7\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) = 0 \]
In conclusion:
\[ \cos\left(\frac{7\pi}{2}\right) = 0 \]