Question

Evaluate : \(\frac{\cos 45^\circ}{\tan 30^\circ + \sin 60^\circ}\)

Answer 1

Evaluate the Expression

Given:

• \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
• \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)
• \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)

Expression:

\[ \frac{\cos 45^\circ}{\tan 30^\circ + \sin 60^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2}} \]

Simplifying the denominator:

\[ \frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2} = \frac{2 + 3}{2\sqrt{3}} = \frac{5}{2\sqrt{3}} \]

Substitute into the main expression:

\[ \frac{\frac{1}{\sqrt{2}}}{\frac{5}{2\sqrt{3}}} = \frac{1}{\sqrt{2}} \cdot \frac{2\sqrt{3}}{5} = \frac{2\sqrt{3}}{5\sqrt{2}} \]

Rationalizing the denominator:

\[ \frac{2\sqrt{3}}{5\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{6}}{10} = \frac{\sqrt{6}}{5} \]

Final Answer:

\[ \boxed{\frac{\sqrt{6}}{5}} \]