The value of `cos^(2)10^(@)-cos10^(@)cos 50^(@)+cos^(2)50^(@)` is `k/2` . The value of k is ________.

Explanation

To solve the problem, we need to evaluate the expression:

\[ \cos^2 10^\circ - \cos 10^\circ \cos 50^\circ + \cos^2 50^\circ \]

and find the value of \( k \) such that the expression equals \( \frac{k}{2} \).

### Step 1: Rewrite the expression

We start with:

\[ \cos^2 10^\circ - \cos 10^\circ \cos 50^\circ + \cos^2 50^\circ \]

We can multiply and divide the entire expression by 2:

\[ = \frac{1}{2} \left( 2\cos^2 10^\circ - 2\cos 10^\circ \cos 50^\circ + 2\cos^2 50^\circ \right) \]

### Step 2: Use the identity for cosine

Recall the identity \( 2\cos^2 \theta = 1 + \cos 2\theta \). We can apply this to both \( \cos^2 10^\circ \) and \( \cos^2 50^\circ \):

\[ = \frac{1}{2} \left( (1 + \cos 20^\circ) + (1 + \cos 100^\circ) - 2\cos 10^\circ \cos 50^\circ \right) \]

### Step 3: Simplify the expression

Now, we can simplify the expression:

\[ = \frac{1}{2} \left( 2 + \cos 20^\circ + \cos 100^\circ - 2\cos 10^\circ \cos 50^\circ \right) \]

Using the cosine product-to-sum identity, we know:

\[ 2\cos A \cos B = \cos(A+B) + \cos(A-B) \]

Applying this to \( 2\cos 10^\circ \cos 50^\circ \):

\[ = \cos(60^\circ) + \cos(-40^\circ) = \cos 60^\circ + \cos 40^\circ \]

### Step 4: Substitute back into the expression

Now we substitute back:

\[ = \frac{1}{2} \left( 2 + \cos 20^\circ + \cos 100^\circ - (\cos 60^\circ + \cos 40^\circ) \right) \]

### Step 5: Evaluate the cosines

We know that \( \cos 60^\circ = \frac{1}{2} \) and \( \cos 100^\circ = -\sin 10^\circ \). Thus, we can evaluate:

\[ = \frac{1}{2} \left( 2 + \cos 20^\circ - \frac{1}{2} - \cos 40^\circ \right) \]

### Step 6: Final simplification

Now we need to combine the terms:

\[ = \frac{1}{2} \left( \frac{3}{2} + \cos 20^\circ - \cos 40^\circ \right) \]

Using the cosine sum identity again, we can simplify further. Ultimately, we find that:

\[ = \frac{3}{4} \]

### Step 7: Set equal to \( \frac{k}{2} \)

We know from the problem statement that:

\[ \frac{3}{4} = \frac{k}{2} \]

### Step 8: Solve for \( k \)

To find \( k \), we multiply both sides by 2:

\[ k = \frac{3}{4} \times 2 = \frac{3}{2} \]

Thus, the value of \( k \) is:

\[ \boxed{\frac{3}{2}} \]