Evaluate `(cos 45^(@))/(sec 30^(@) + cos 30^(@))`

Answer

To evaluate the expression \(\frac{\cos 45^\circ}{\sec 30^\circ + \cos 30^\circ}\), we will follow these steps:

### Step 1: Find the values of \(\cos 45^\circ\), \(\sec 30^\circ\), and \(\cos 30^\circ\).

- The value of \(\cos 45^\circ\) is \(\frac{1}{\sqrt{2}}\).
- The value of \(\sec 30^\circ\) is \(\frac{1}{\cos 30^\circ}\). Since \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), we have:
\[ \sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}. \]
- The value of \(\cos 30^\circ\) is \(\frac{\sqrt{3}}{2}\).

### Step 2: Substitute the values into the expression.

Now substituting these values into the expression:
\[ \frac{\cos 45^\circ}{\sec 30^\circ + \cos 30^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{2}}. \]

### Step 3: Simplify the denominator.

To simplify the denominator \(\frac{2}{\sqrt{3}} + \frac{\sqrt{3}}{2}\), we need a common denominator. The common denominator of \(\sqrt{3}\) and \(2\) is \(2\sqrt{3}\).

Rewriting each term:
\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}}, \]
\[ \frac{\sqrt{3}}{2} = \frac{\sqrt{3} \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{3}{2\sqrt{3}}. \]

Now adding them together:
\[ \frac{4}{2\sqrt{3}} + \frac{3}{2\sqrt{3}} = \frac{4 + 3}{2\sqrt{3}} = \frac{7}{2\sqrt{3}}. \]

### Step 4: Substitute back into the expression.

Now substituting this back into the expression, we have:
\[ \frac{\frac{1}{\sqrt{2}}}{\frac{7}{2\sqrt{3}}} = \frac{1}{\sqrt{2}} \cdot \frac{2\sqrt{3}}{7} = \frac{2\sqrt{3}}{7\sqrt{2}}. \]

### Step 5: Rationalize the denominator.

To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{2}\):
\[ \frac{2\sqrt{3} \cdot \sqrt{2}}{7\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{6}}{14} = \frac{\sqrt{6}}{7}. \]

### Final Answer:
Thus, the value of the expression \(\frac{\cos 45^\circ}{\sec 30^\circ + \cos 30^\circ}\) is \(\frac{\sqrt{6}}{7}\).

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