Find the exact value (a) \( \cot 20^{\circ}-\frac{\sin 70^{\circ}}{\sin 20^{\circ}} \)

To find the exact value of the expression: \[ \cot 20^{\circ} - \frac{\sin 70^{\circ}}{\sin 20^{\circ}} \] Let's break it down step by step. 1. **Express \(\cot 20^{\circ}\) in terms of sine and cosine:** \[ \cot 20^{\circ} = \frac{\cos 20^{\circ}}{\sin 20^{\circ}} \] 2. **Recognize that \(\sin 70^{\circ}\) is equivalent to \(\cos 20^{\circ}\):** \[ \sin 70^{\circ} = \sin (90^{\circ} - 20^{\circ}) = \cos 20^{\circ} \] 3. **Substitute \(\sin 70^{\circ}\) with \(\cos 20^{\circ}\) in the original expression:** \[ \cot 20^{\circ} - \frac{\sin 70^{\circ}}{\sin 20^{\circ}} = \frac{\cos 20^{\circ}}{\sin 20^{\circ}} - \frac{\cos 20^{\circ}}{\sin 20^{\circ}} = 0 \] Thus, the exact value of the expression is: **Answer:** 0