What is the value of cos `(30^@+x) cos (30^@-x)- sin (30^@+x) sin (30^@-x)`

Answer

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

\[ \cos(30^\circ + x) \cos(30^\circ - x) - \sin(30^\circ + x) \sin(30^\circ - x) \]

We can use the cosine addition formula, which states that:

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

### Step 1: Identify A and B
In our case, we can identify:
- \( A = 30^\circ + x \)
- \( B = 30^\circ - x \)

### Step 2: Calculate A + B
Now, we calculate \( A + B \):

\[ A + B = (30^\circ + x) + (30^\circ - x) = 30^\circ + x + 30^\circ - x = 60^\circ \]

### Step 3: Apply the Cosine Addition Formula
Using the cosine addition formula, we have:

\[ \cos(30^\circ + x) \cos(30^\circ - x) - \sin(30^\circ + x) \sin(30^\circ - x) = \cos(60^\circ) \]

### Step 4: Calculate cos(60°)
Now, we know that:

\[ \cos(60^\circ) = \frac{1}{2} \]

### Final Result
Thus, the value of the original expression is:

\[ \cos(30^\circ + x) \cos(30^\circ - x) - \sin(30^\circ + x) \sin(30^\circ - x) = \frac{1}{2} \]

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