The value of `cos^2 5^@ + cos^2 10^@ + cos^2 15^@ + cos^2 20^@+ cos^2 70^@ + cos^2 75^@+ cos^2 80^@ + cos^2 85^@` is equal to

Answer

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

\[ \cos^2 5^\circ + \cos^2 10^\circ + \cos^2 15^\circ + \cos^2 20^\circ + \cos^2 70^\circ + \cos^2 75^\circ + \cos^2 80^\circ + \cos^2 85^\circ \]

### Step 1: Use the identity for cosine

We can use the identity \(\cos^2 \theta = \frac{1 + \cos 2\theta}{2}\) to rewrite each term in the expression.

### Step 2: Rewrite each cosine term

Applying the identity:

\[ \cos^2 5^\circ = \frac{1 + \cos 10^\circ}{2} \]
\[ \cos^2 10^\circ = \frac{1 + \cos 20^\circ}{2} \]
\[ \cos^2 15^\circ = \frac{1 + \cos 30^\circ}{2} \]
\[ \cos^2 20^\circ = \frac{1 + \cos 40^\circ}{2} \]
\[ \cos^2 70^\circ = \frac{1 + \cos 140^\circ}{2} \]
\[ \cos^2 75^\circ = \frac{1 + \cos 150^\circ}{2} \]
\[ \cos^2 80^\circ = \frac{1 + \cos 160^\circ}{2} \]
\[ \cos^2 85^\circ = \frac{1 + \cos 170^\circ}{2} \]

### Step 3: Substitute back into the expression

Now substituting these back into the expression:

\[ \frac{1 + \cos 10^\circ}{2} + \frac{1 + \cos 20^\circ}{2} + \frac{1 + \cos 30^\circ}{2} + \frac{1 + \cos 40^\circ}{2} + \frac{1 + \cos 140^\circ}{2} + \frac{1 + \cos 150^\circ}{2} + \frac{1 + \cos 160^\circ}{2} + \frac{1 + \cos 170^\circ}{2} \]

### Step 4: Combine the terms

Combining all the terms gives:

\[ \frac{8}{2} + \frac{1}{2}(\cos 10^\circ + \cos 20^\circ + \cos 30^\circ + \cos 40^\circ + \cos 140^\circ + \cos 150^\circ + \cos 160^\circ + \cos 170^\circ) \]

### Step 5: Simplify the cosine terms

Notice that \(\cos 140^\circ = -\cos 40^\circ\), \(\cos 150^\circ = -\cos 30^\circ\), \(\cos 160^\circ = -\cos 20^\circ\), and \(\cos 170^\circ = -\cos 10^\circ\). Thus, the sum of the cosines becomes:

\[ \cos 10^\circ + \cos 20^\circ + \cos 30^\circ + \cos 40^\circ - \cos 40^\circ - \cos 30^\circ - \cos 20^\circ - \cos 10^\circ = 0 \]

### Step 6: Final calculation

So, the entire expression simplifies to:

\[ \frac{8}{2} + 0 = 4 \]

### Conclusion

Thus, the value of the expression is:

\[ \boxed{4} \]