`("cos"^(4)(pi)/(24)-"sin"^(4)(pi)/(24))` equals :

Options

A. `(1)/(sqrt(2))`
B. `(sqrt(6)-sqrt(2))/(4)`
C. `(sqrt(6)+sqrt(2))/(4)`
D. `(sqrt(3)+1)/(2)`

Answer

To solve the expression \( \cos^4\left(\frac{\pi}{24}\right) - \sin^4\left(\frac{\pi}{24}\right) \), we can use the difference of squares identity. Here’s the step-by-step solution:

### Step 1: Recognize the difference of squares
The expression can be rewritten using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ \cos^4\left(\frac{\pi}{24}\right) - \sin^4\left(\frac{\pi}{24}\right) = \left(\cos^2\left(\frac{\pi}{24}\right) - \sin^2\left(\frac{\pi}{24}\right)\right)\left(\cos^2\left(\frac{\pi}{24}\right) + \sin^2\left(\frac{\pi}{24}\right)\right) \]

### Step 2: Simplify using the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \cos^2\theta + \sin^2\theta = 1 \]
Thus, we can simplify the expression:
\[ \cos^2\left(\frac{\pi}{24}\right) + \sin^2\left(\frac{\pi}{24}\right) = 1 \]
So, our expression simplifies to:
\[ \cos^4\left(\frac{\pi}{24}\right) - \sin^4\left(\frac{\pi}{24}\right) = \left(\cos^2\left(\frac{\pi}{24}\right) - \sin^2\left(\frac{\pi}{24}\right)\right)(1) \]
This simplifies to:
\[ \cos^2\left(\frac{\pi}{24}\right) - \sin^2\left(\frac{\pi}{24}\right) \]

### Step 3: Use the double angle identity
We can use the double angle identity for cosine, which states:
\[ \cos(2\theta) = \cos^2\theta - \sin^2\theta \]
In our case, we have:
\[ \cos^2\left(\frac{\pi}{24}\right) - \sin^2\left(\frac{\pi}{24}\right) = \cos\left(2 \cdot \frac{\pi}{24}\right) = \cos\left(\frac{\pi}{12}\right) \]

### Step 4: Find the value of \( \cos\left(\frac{\pi}{12}\right) \)
To find \( \cos\left(\frac{\pi}{12}\right) \), we can express \( \frac{\pi}{12} \) as \( \frac{\pi}{4} - \frac{\pi}{6} \). Using the cosine subtraction formula:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
Let \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{6} \):
\[ \cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right) \]

### Step 5: Substitute known values
Now substituting the known values:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, we have:
\[ \cos\left(\frac{\pi}{12}\right) = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) \]
This simplifies to:
\[ \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 6: Final simplification
To express this in a more standard form, we multiply the numerator and denominator by \( \sqrt{2} \):
\[ \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2}(\sqrt{3} + 1)}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

### Conclusion
Thus, the final answer is:
\[ \cos^4\left(\frac{\pi}{24}\right) - \sin^4\left(\frac{\pi}{24}\right) = \frac{\sqrt{6} + \sqrt{2}}{4} \]