The value of `cos^(2)75^(@)+cos^(2)45^(@)+cos^(2)15^(@)-cos^(2)30^(@)-cos^(2)60^(@)` is

Options

A. 0
B. 1
C. `1//2`
D. `1//4`

Answer

To solve the expression \( \cos^2 75^\circ + \cos^2 45^\circ + \cos^2 15^\circ - \cos^2 30^\circ - \cos^2 60^\circ \), we will compute each cosine value and then substitute them into the equation.

### Step 1: Calculate \( \cos^2 75^\circ \)
Using the identity \( \cos(90^\circ - \theta) = \sin(\theta) \):
\[ \cos 75^\circ = \sin 15^\circ \]
Thus,
\[ \cos^2 75^\circ = \sin^2 15^\circ \]

### Step 2: Calculate \( \cos^2 45^\circ \)
The value of \( \cos 45^\circ \) is:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 3: Calculate \( \cos^2 15^\circ \)
We will keep \( \cos^2 15^\circ \) as it is for now.

### Step 4: Calculate \( \cos^2 30^\circ \)
The value of \( \cos 30^\circ \) is:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \quad \Rightarrow \quad \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 5: Calculate \( \cos^2 60^\circ \)
The value of \( \cos 60^\circ \) is:
\[ \cos 60^\circ = \frac{1}{2} \quad \Rightarrow \quad \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 6: Substitute all values into the expression
Now substituting back into the expression:
\[ \sin^2 15^\circ + \frac{1}{2} + \cos^2 15^\circ - \frac{3}{4} - \frac{1}{4} \]

### Step 7: Simplify the expression
Notice that \( \sin^2 15^\circ + \cos^2 15^\circ = 1 \):
\[ 1 + \frac{1}{2} - \frac{3}{4} - \frac{1}{4} = 1 + \frac{1}{2} - 1 = \frac{1}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{2} \]