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: Use the complementary angle identities
Recall that:
- \( \tan(90^\circ - x) = \cot x \)
- \( \cot(90^\circ - x) = \tan x \)

Using these identities, we can simplify the terms in the expression.

### Step 2: Simplify \( \cot 37^\circ \)
Since \( \cot 37^\circ = \tan(90^\circ - 37^\circ) = \tan 53^\circ \), we can rewrite the first term:
\[ \frac{2 \tan 53^\circ}{\cot 37^\circ} = \frac{2 \tan 53^\circ}{\tan 53^\circ} = 2 \]

### Step 3: Simplify \( \cot 80^\circ \)
Similarly, for the second term, we have:
\[ \cot 80^\circ = \tan(90^\circ - 80^\circ) = \tan 10^\circ \]
Thus, we can rewrite the second term:
\[ \frac{\cot 80^\circ}{\tan 10^\circ} = \frac{\tan 10^\circ}{\tan 10^\circ} = 1 \]

### Step 4: Combine the results
Now substituting back into the original expression, we get:
\[ 2 - 1 = 1 \]

### Final Answer
The evaluated expression is \( 1 \).

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