3. \( \csc 60^{\circ}-\tan 45^{\circ}+\cos 60^{\circ} \)

To evaluate \( \csc 60^{\circ}-\tan 45^{\circ}+\cos 60^{\circ} \), let's break it down using known trigonometric values. - The cosecant of \( 60^{\circ} \) is \( \csc 60^{\circ} = \frac{1}{\sin 60^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \). - The tangent of \( 45^{\circ} \) is \( \tan 45^{\circ} = 1 \). - The cosine of \( 60^{\circ} \) is \( \cos 60^{\circ} = \frac{1}{2} \). Putting this all together gives: \[ \frac{2}{\sqrt{3}} - 1 + \frac{1}{2} \] Now, we can simplify: 1. Convert all terms to have a common denominator, which is \( 2\sqrt{3} \): - \( \frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}} \) - \( 1 = \frac{2\sqrt{3}}{2\sqrt{3}} \) - \( \frac{1}{2} = \frac{\sqrt{3}}{2\sqrt{3}} \) Then combine them: \[ \frac{4 - 2\sqrt{3} + \sqrt{3}}{2\sqrt{3}} = \frac{4 - \sqrt{3}}{2\sqrt{3}} \] Thus, the final answer simplifies to: \[ \frac{4 - \sqrt{3}}{2\sqrt{3}} \] This is \( \csc 60^{\circ}-\tan 45^{\circ}+\cos 60^{\circ} \) evaluated and simplified!