What is the value of `(cos15^(@)+cos45^(@))/(cos^(3)15^(@)+cos^(3)45^(@))`

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(1)/(3)`
D. None of these

Explanation

To solve the expression \((\cos 15^\circ + \cos 45^\circ) / (\cos^3 15^\circ + \cos^3 45^\circ)\), we will follow these steps:

### Step 1: Calculate \(\cos 15^\circ\) and \(\cos 45^\circ\)

We know:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \text{(exact value)} \]
To find \(\cos 15^\circ\), we can use the cosine subtraction formula:
\[ \cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \]
Substituting the known values:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Thus,
\[ \cos 15^\circ = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 2: Substitute values into the expression

Now we substitute \(\cos 15^\circ\) and \(\cos 45^\circ\) into the expression:
\[ \cos 15^\circ + \cos 45^\circ = \frac{\sqrt{3} + 1}{2\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{\sqrt{3} + 1 + 2}{2\sqrt{2}} = \frac{\sqrt{3} + 3}{2\sqrt{2}} \]

### Step 3: Calculate \(\cos^3 15^\circ\) and \(\cos^3 45^\circ\)

Using the values calculated:
\[ \cos^3 15^\circ = \left(\frac{\sqrt{3} + 1}{2\sqrt{2}}\right)^3 = \frac{(\sqrt{3} + 1)^3}{8 \cdot 2\sqrt{2}} = \frac{(\sqrt{3} + 1)^3}{16\sqrt{2}} \]
\[ \cos^3 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^3 = \frac{1}{2\sqrt{2}} \]

### Step 4: Substitute into the denominator

Now we substitute these values into the denominator:
\[ \cos^3 15^\circ + \cos^3 45^\circ = \frac{(\sqrt{3} + 1)^3}{16\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{(\sqrt{3} + 1)^3 + 8}{16\sqrt{2}} \]

### Step 5: Combine the results

Now we can substitute the results into the original expression:
\[ \frac{\frac{\sqrt{3} + 3}{2\sqrt{2}}}{\frac{(\sqrt{3} + 1)^3 + 8}{16\sqrt{2}}} \]
This simplifies to:
\[ \frac{16(\sqrt{3} + 3)}{2((\sqrt{3} + 1)^3 + 8)} = \frac{8(\sqrt{3} + 3)}{(\sqrt{3} + 1)^3 + 8} \]

### Step 6: Simplify further

Now we need to simplify \((\sqrt{3} + 1)^3 + 8\):
\[ (\sqrt{3} + 1)^3 = 3\sqrt{3} + 3 + 3\sqrt{3} + 1 = 4 + 6\sqrt{3} \]
Thus,
\[ (\sqrt{3} + 1)^3 + 8 = 4 + 6\sqrt{3} + 8 = 12 + 6\sqrt{3} \]

### Final Result

The final expression is:
\[ \frac{8(\sqrt{3} + 3)}{12 + 6\sqrt{3}} \]
This can be simplified further, but we can check if it matches any of the options provided.