Evaluate : `(cos^(2) 20° + cos^(2) 70°) + (cot 25^(@))/( tan 65^(@))` + cot 5° cot 10° cot 60° cot 80° cot 85°.

Answer

To evaluate the expression \((\cos^2 20° + \cos^2 70°) + \frac{\cot 25°}{\tan 65°} + \cot 5° \cot 10° \cot 60° \cot 80° \cot 85°\), we will simplify each part step by step.

### Step 1: Simplifying \(\cos^2 20° + \cos^2 70°\)

Using the identity \(\cos(90° - \theta) = \sin \theta\), we can express \(\cos^2 70°\) as:

\[ \cos^2 70° = \sin^2 20° \]

Thus, we can rewrite the expression:

\[ \cos^2 20° + \cos^2 70° = \cos^2 20° + \sin^2 20° \]

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ \cos^2 20° + \sin^2 20° = 1 \]

### Step 2: Simplifying \(\frac{\cot 25°}{\tan 65°}\)

Using the identity \(\tan(90° - \theta) = \cot \theta\), we can express \(\tan 65°\) as:

\[ \tan 65° = \cot(90° - 65°) = \cot 25° \]

Thus, we have:

\[ \frac{\cot 25°}{\tan 65°} = \frac{\cot 25°}{\cot 25°} = 1 \]

### Step 3: Simplifying \(\cot 5° \cot 10° \cot 60° \cot 80° \cot 85°\)

Using the identity \(\cot(90° - \theta) = \tan \theta\), we can pair the cotangents:

\[ \cot 85° = \tan 5° \quad \text{and} \quad \cot 80° = \tan 10° \]

Thus, we can rewrite:

\[ \cot 5° \cot 85° = \cot 5° \tan 5° = 1 \]

\[ \cot 10° \cot 80° = \cot 10° \tan 10° = 1 \]

Now we have:

\[ \cot 5° \cot 10° \cot 60° \cot 80° \cot 85° = 1 \cdot 1 \cdot \cot 60° \]

### Step 4: Evaluating \(\cot 60°\)

The value of \(\cot 60°\) is:

\[ \cot 60° = \frac{1}{\tan 60°} = \frac{1}{\sqrt{3}} \]

### Step 5: Putting it all together

Now we can combine all parts:

\[ 1 + 1 + \cot 60° = 1 + 1 + \frac{1}{\sqrt{3}} = 2 + \frac{1}{\sqrt{3}} \]

### Step 6: Rationalizing the expression

To express \(2 + \frac{1}{\sqrt{3}}\) in a standard form, we can rationalize:

\[ 2 + \frac{1}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{2\sqrt{3} + 1}{\sqrt{3}} \]

To eliminate the square root in the denominator, multiply the numerator and denominator by \(\sqrt{3}\):

\[ \frac{(2\sqrt{3} + 1)\sqrt{3}}{3} = \frac{2 \cdot 3 + \sqrt{3}}{3} = \frac{6 + \sqrt{3}}{3} \]

### Final Answer

Thus, the final evaluated expression is:

\[ \frac{6 + \sqrt{3}}{3} \]