Find the value of `cos ^(2) 45 ^(@) - sin ^(2) 15 ^(@).`

Answer

To solve the problem of finding the value of \( \cos^2 45^\circ - \sin^2 15^\circ \), we can follow these steps:

### Step 1: Calculate \( \cos^2 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]
Thus,
\[ \cos^2 45^\circ = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} \]

### Step 2: Calculate \( \sin^2 15^\circ \)
To find \( \sin^2 15^\circ \), we can use the identity:
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
First, we need to find \( \cos 30^\circ \) because \( 15^\circ \) can be expressed as \( \frac{30^\circ}{2} \):
\[ \sin^2 15^\circ = 1 - \cos^2 15^\circ \]
Using the double angle formula for cosine:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^2 15^\circ = \frac{1 + \cos 30^\circ}{2} = \frac{1 + \frac{\sqrt{3}}{2}}{2} = \frac{2 + \sqrt{3}}{4} \]
Now substituting back:
\[ \sin^2 15^\circ = 1 - \frac{2 + \sqrt{3}}{4} = \frac{4 - (2 + \sqrt{3})}{4} = \frac{2 - \sqrt{3}}{4} \]

### Step 3: Substitute the values into the expression
Now we can substitute the values we found into the original expression:
\[ \cos^2 45^\circ - \sin^2 15^\circ = \frac{1}{2} - \frac{2 - \sqrt{3}}{4} \]
Finding a common denominator (which is 4):
\[ \frac{1}{2} = \frac{2}{4} \]
Thus,
\[ \cos^2 45^\circ - \sin^2 15^\circ = \frac{2}{4} - \frac{2 - \sqrt{3}}{4} = \frac{2 - (2 - \sqrt{3})}{4} = \frac{\sqrt{3}}{4} \]

### Final Answer
The value of \( \cos^2 45^\circ - \sin^2 15^\circ \) is:
\[ \frac{\sqrt{3}}{4} \]