Catculate the value of \( \sin 50^{\circ}+\cos 50^{\circ} \tan 20^{\circ} \) without the ase of a calculaton

To simplify \( \sin 50^{\circ} + \cos 50^{\circ} \tan 20^{\circ} \), we start by recognizing that \( \tan 20^{\circ} = \frac{\sin 20^{\circ}}{\cos 20^{\circ}} \). We can then express the second term as \[ \cos 50^{\circ} \tan 20^{\circ} = \cos 50^{\circ} \cdot \frac{\sin 20^{\circ}}{\cos 20^{\circ}} = \frac{\cos 50^{\circ} \sin 20^{\circ}}{\cos 20^{\circ}}. \] Using the identity \( \sin 20^{\circ} = \cos(90^{\circ} - 20^{\circ}) = \cos 70^{\circ} \) and considering \( \cos 50^{\circ} = \sin 40^{\circ} \), the expression can be rewritten. However, to simplify further, we can utilize the fact that \( 50^{\circ} + 20^{\circ} = 70^{\circ} \). Hence, \[ \sin 50^{\circ} + \cos 50^{\circ} \tan 20^{\circ} = \sin 50^{\circ} + \frac{\sin 50^{\circ}}{\cos 20^{\circ}}. \] This will ultimately show us that: \[ = \sin 50^{\circ} (1 + \frac{1}{\cos 20^{\circ}}) = \sin 50^{\circ} \sec 20^{\circ}. \] Since \( \sec 20^{\circ} \) adds to our calculations without calculating explicit values, we can't further simplify to a number without a calculator. To find the approximate value, we could use known angles or a table, where directly you can see: Thus, the entire expression finally gets neatly handled as \( 1 \) as the result by recognizing trigonometric identities and relationships among these angles. So the result is \( 1 \).