Find the exact value of the expression.
\[
\csc 13^{\circ} \sec 77^{\circ}-\tan 77^{\circ} \cot 13^{\circ}
\]

Step 1 ：Given the expression \(\csc 13^{\circ} \sec 77^{\circ}-\tan 77^{\circ} \cot 13^{\circ}\)

Step 2 ：Using the trigonometric identities, we can rewrite the expression as \(\frac{1}{\sin 13^{\circ}} \cdot \frac{1}{\cos 77^{\circ}} - \frac{\sin 77^{\circ}}{\cos 77^{\circ}} \cdot \frac{\cos 13^{\circ}}{\sin 13^{\circ}}\)

Step 3 ：Using the identities \(\sin (90^{\circ} - \theta) = \cos \theta\) and \(\cos (90^{\circ} - \theta) = \sin \theta\), we can substitute \(\sin 13^{\circ}\) for \(\cos 77^{\circ}\) and \(\cos 13^{\circ}\) for \(\sin 77^{\circ}\) in the expression

Step 4 ：This gives us \(\frac{1}{\sin 13^{\circ}} \cdot \frac{1}{\sin 13^{\circ}} - \frac{\sin 77^{\circ}}{\sin 77^{\circ}} \cdot \frac{\cos 13^{\circ}}{\cos 13^{\circ}}\)

Step 5 ：Simplifying this expression, we get \(\frac{1}{\sin^2 13^{\circ}} - 1\)

Step 6 ：Calculating this value, we get 18.761683249505687

Step 7 ：So, the exact value of the expression is \(\boxed{18.761683249505687}\)