Find the valueof `2cos45^(@).cos15^(@)`:

Options

A. `((sqrt3+1))/(2sqrt2)`
B. `((sqrt3-1))/(2)`
C. `(sqrt3+1)/(2)`
D. none of these

Answer

To find the value of \( 2 \cos 45^\circ \cdot \cos 15^\circ \), we can follow these steps:

### Step 1: Substitute the value of \( \cos 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]
So we can substitute this value into our expression:
\[ 2 \cos 45^\circ \cdot \cos 15^\circ = 2 \cdot \frac{1}{\sqrt{2}} \cdot \cos 15^\circ \]

### Step 2: Simplify the expression
Now, we can simplify the expression:
\[ = \frac{2}{\sqrt{2}} \cdot \cos 15^\circ = \sqrt{2} \cdot \cos 15^\circ \]

### Step 3: Use the cosine angle subtraction formula
We can express \( \cos 15^\circ \) using the cosine angle 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:
\[ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]

### Step 4: Calculate the individual terms
Calculating each term:
\[ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} \]

### Step 5: Combine the terms
Now, we can combine the two fractions:
\[ = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 6: Substitute back into the expression
Now, substitute back into our expression for \( 2 \cos 45^\circ \cdot \cos 15^\circ \):
\[ \sqrt{2} \cdot \cos 15^\circ = \sqrt{2} \cdot \frac{\sqrt{3} + 1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2} \]

### Final Result
Thus, the value of \( 2 \cos 45^\circ \cdot \cos 15^\circ \) is:
\[ \frac{\sqrt{3} + 1}{2} \]