The value of `(cos^4 1^@+cos^4 2^@+....+cos^4 179^@)-(sin^4 1^@+sin^4 2^@+.....+sin^4 179^@)` equals

Explanation

To solve the problem, we need to find the value of the expression:

\[ S = (\cos^4 1^\circ + \cos^4 2^\circ + \ldots + \cos^4 179^\circ) - (\sin^4 1^\circ + \sin^4 2^\circ + \ldots + \sin^4 179^\circ) \]

### Step 1: Rewrite the Expression

We can rewrite \( S \) as:

\[ S = \sum_{k=1}^{179} \cos^4 k^\circ - \sum_{k=1}^{179} \sin^4 k^\circ \]

### Step 2: Combine Terms

We can combine the sums:

\[ S = \sum_{k=1}^{179} (\cos^4 k^\circ - \sin^4 k^\circ) \]

### Step 3: Use the Identity

Using the identity \( a^4 - b^4 = (a^2 + b^2)(a^2 - b^2) \), we can express \( \cos^4 k^\circ - \sin^4 k^\circ \) as:

\[ \cos^4 k^\circ - \sin^4 k^\circ = (\cos^2 k^\circ + \sin^2 k^\circ)(\cos^2 k^\circ - \sin^2 k^\circ) \]

Since \( \cos^2 k^\circ + \sin^2 k^\circ = 1 \), we have:

\[ \cos^4 k^\circ - \sin^4 k^\circ = \cos^2 k^\circ - \sin^2 k^\circ \]

### Step 4: Substitute Back

Now substitute this back into the expression for \( S \):

\[ S = \sum_{k=1}^{179} (\cos^2 k^\circ - \sin^2 k^\circ) \]

### Step 5: Use the Cosine Double Angle Identity

Using the identity \( \cos^2 k^\circ - \sin^2 k^\circ = \cos(2k^\circ) \):

\[ S = \sum_{k=1}^{179} \cos(2k^\circ) \]

### Step 6: Evaluate the Sum

The sum \( \sum_{k=1}^{n} \cos(2k^\circ) \) can be evaluated using the formula for the sum of cosines:

\[ \sum_{k=0}^{n-1} \cos(a + kd) = \frac{\sin\left(\frac{nd}{2}\right) \cos\left(a + \frac{(n-1)d}{2}\right)}{\sin\left(\frac{d}{2}\right)} \]

Here, \( a = 2^\circ \), \( d = 2^\circ \), and \( n = 179 \):

\[ S = \frac{\sin\left(179 \cdot 2^\circ / 2\right) \cos\left(2^\circ + \frac{(179-1) \cdot 2^\circ}{2}\right)}{\sin(1^\circ)} \]

### Step 7: Simplify

Calculating \( S \):

\[ S = \frac{\sin(179^\circ) \cos(180^\circ)}{\sin(1^\circ)} \]

Since \( \sin(179^\circ) = \sin(180^\circ - 1^\circ) = \sin(1^\circ) \) and \( \cos(180^\circ) = -1 \):

\[ S = \frac{\sin(1^\circ)(-1)}{\sin(1^\circ)} = -1 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{-1} \]