Given $\sin\alpha = \dfrac{144}{145}$ and $\tan\alpha = \dfrac{144}{17}$.

Use the identity $\tan\alpha = \dfrac{\sin\alpha}{\cos\alpha}$, which gives:

$\cos\alpha = \frac{\sin\alpha}{\tan\alpha}$

Substitute the known values:

$\cos\alpha = \frac{\dfrac{144}{145}}{\dfrac{144}{17}} = \frac{144}{145} \times \frac{17}{144} = \frac{17}{145}$