What is the value of `cos15^(@)-cos165^(@)` ?

Options

A. `(sqrt(3))/(sqrt(2))`
B. `(2)/(sqrt(3)-1)`
C. `(sqrt(3)+1)/(sqrt(2))`
D. `((sqrt(3)+1))/(2)`

Answer

To find the value of \( \cos 15^\circ - \cos 165^\circ \), we can follow these steps:

### Step 1: Rewrite \( \cos 165^\circ \)
We know that \( \cos(180^\circ - \theta) = -\cos(\theta) \). Therefore, we can rewrite \( \cos 165^\circ \) as:
\[ \cos 165^\circ = \cos(180^\circ - 15^\circ) = -\cos 15^\circ \]

### Step 2: Substitute in the expression
Now we can substitute this back into our original expression:
\[ \cos 15^\circ - \cos 165^\circ = \cos 15^\circ - (-\cos 15^\circ) \]
This simplifies to:
\[ \cos 15^\circ + \cos 15^\circ = 2\cos 15^\circ \]

### Step 3: Use the cosine subtraction formula
Next, we can express \( \cos 15^\circ \) in terms of known angles. We can use the cosine subtraction formula:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
We can write \( 15^\circ \) as \( 60^\circ - 45^\circ \):
\[ \cos 15^\circ = \cos(60^\circ - 45^\circ) \]
Using the formula:
\[ \cos 15^\circ = \cos 60^\circ \cos 45^\circ + \sin 60^\circ \sin 45^\circ \]

### Step 4: Substitute known values
Now we substitute the known values:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)

Substituting these values gives:
\[ \cos 15^\circ = \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) \]
This simplifies to:
\[ \cos 15^\circ = \frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}} = \frac{1 + \sqrt{3}}{2\sqrt{2}} \]

### Step 5: Final calculation
Now, substituting back into our expression for \( 2\cos 15^\circ \):
\[ 2\cos 15^\circ = 2 \times \frac{1 + \sqrt{3}}{2\sqrt{2}} = \frac{1 + \sqrt{3}}{\sqrt{2}} \]

### Final Answer
Thus, the value of \( \cos 15^\circ - \cos 165^\circ \) is:
\[ \frac{1 + \sqrt{3}}{\sqrt{2}} \]