Find the value of `(cot 25^@+cot55^@)/(tan 25^@ + tan 55^@)+(cot 55^@ + cot100^@)/(tan 55^@ + tan 100^@)+(cot100^@ + cot 25^@)/(tan 100^@+tan 25^@)`

Answer

To find the value of the expression

\[ \frac{\cot 25^\circ + \cot 55^\circ}{\tan 25^\circ + \tan 55^\circ} + \frac{\cot 55^\circ + \cot 100^\circ}{\tan 55^\circ + \tan 100^\circ} + \frac{\cot 100^\circ + \cot 25^\circ}{\tan 100^\circ + \tan 25^\circ} \]

we will simplify each term step by step.

### Step 1: Simplifying the first term

We start with the first term:

\[ \frac{\cot 25^\circ + \cot 55^\circ}{\tan 25^\circ + \tan 55^\circ} \]

Using the identity \(\cot x = \frac{1}{\tan x}\), we can rewrite the numerator:

\[ \cot 25^\circ + \cot 55^\circ = \frac{1}{\tan 25^\circ} + \frac{1}{\tan 55^\circ} = \frac{\tan 55^\circ + \tan 25^\circ}{\tan 25^\circ \tan 55^\circ} \]

Now substituting this back into the expression:

\[ \frac{\frac{\tan 55^\circ + \tan 25^\circ}{\tan 25^\circ \tan 55^\circ}}{\tan 25^\circ + \tan 55^\circ} \]

This simplifies to:

\[ \frac{1}{\tan 25^\circ \tan 55^\circ} \]

### Step 2: Simplifying the second term

Now, consider the second term:

\[ \frac{\cot 55^\circ + \cot 100^\circ}{\tan 55^\circ + \tan 100^\circ} \]

Again using the cotangent identity:

\[ \cot 55^\circ + \cot 100^\circ = \frac{1}{\tan 55^\circ} + \frac{1}{\tan 100^\circ} = \frac{\tan 100^\circ + \tan 55^\circ}{\tan 55^\circ \tan 100^\circ} \]

Substituting this back gives:

\[ \frac{\frac{\tan 100^\circ + \tan 55^\circ}{\tan 55^\circ \tan 100^\circ}}{\tan 55^\circ + \tan 100^\circ} \]

This simplifies to:

\[ \frac{1}{\tan 55^\circ \tan 100^\circ} \]

### Step 3: Simplifying the third term

Now, we simplify the third term:

\[ \frac{\cot 100^\circ + \cot 25^\circ}{\tan 100^\circ + \tan 25^\circ} \]

Using the cotangent identity again:

\[ \cot 100^\circ + \cot 25^\circ = \frac{1}{\tan 100^\circ} + \frac{1}{\tan 25^\circ} = \frac{\tan 25^\circ + \tan 100^\circ}{\tan 100^\circ \tan 25^\circ} \]

Substituting this back gives:

\[ \frac{\frac{\tan 25^\circ + \tan 100^\circ}{\tan 100^\circ \tan 25^\circ}}{\tan 100^\circ + \tan 25^\circ} \]

This simplifies to:

\[ \frac{1}{\tan 100^\circ \tan 25^\circ} \]

### Step 4: Combining all terms

Now we can combine all three simplified terms:

\[ \frac{1}{\tan 25^\circ \tan 55^\circ} + \frac{1}{\tan 55^\circ \tan 100^\circ} + \frac{1}{\tan 100^\circ \tan 25^\circ} \]

Let \(x = \tan 25^\circ\), \(y = \tan 55^\circ\), and \(z = \tan 100^\circ\). We know from the tangent addition formula that:

\[ \tan 25^\circ + \tan 55^\circ + \tan 100^\circ = \tan 180^\circ = 0 \]

Thus, we can express the sum as:

\[ \frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} \]

This can be rewritten as:

\[ \frac{z + x + y}{xyz} \]

Since \(x + y + z = 0\), we find that:

\[ \frac{0}{xyz} = 0 \]

### Final Result

Thus, the value of the entire expression is:

\[ \boxed{1} \]