Evaluate: `(2tan 53^(@))/(cot 37^(@))-(cot 80^(@))/(tan 10^(@))`

Answer

To evaluate the expression \( \frac{2 \tan 53^\circ}{\cot 37^\circ} - \frac{\cot 80^\circ}{\tan 10^\circ} \), we can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can rewrite \( \cot 37^\circ \) and \( \cot 80^\circ \) as:
\[ \cot 37^\circ = \frac{1}{\tan 37^\circ} \quad \text{and} \quad \cot 80^\circ = \frac{1}{\tan 80^\circ} \]

### Step 2: Substitute cotangent in the expression
Substituting these into the expression gives us:
\[ \frac{2 \tan 53^\circ}{\frac{1}{\tan 37^\circ}} - \frac{\frac{1}{\tan 80^\circ}}{\tan 10^\circ} \]
This simplifies to:
\[ 2 \tan 53^\circ \tan 37^\circ - \frac{1}{\tan 80^\circ \tan 10^\circ} \]

### Step 3: Use the complementary angle identity
We know that:
\[ \tan(90^\circ - \theta) = \cot \theta \]
Thus:
\[ \tan 53^\circ = \cot 37^\circ \quad \text{and} \quad \tan 80^\circ = \cot 10^\circ \]

### Step 4: Substitute the identities back into the expression
Substituting these identities back, we have:
\[ 2 \cot 37^\circ \tan 37^\circ - \frac{1}{\cot 10^\circ \tan 10^\circ} \]
Since \( \cot 37^\circ \tan 37^\circ = 1 \), the first term simplifies to:
\[ 2 \cdot 1 - \frac{1}{\cot 10^\circ \tan 10^\circ} \]

### Step 5: Simplify the second term
Using the identity \( \cot 10^\circ = \frac{1}{\tan 10^\circ} \), we can rewrite the second term:
\[ \frac{1}{\cot 10^\circ \tan 10^\circ} = \frac{1}{\frac{1}{\tan 10^\circ} \tan 10^\circ} = 1 \]

### Step 6: Combine the results
Now we combine the results:
\[ 2 - 1 = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]