Evaluate :
`2 (tan 57^(@))/(cot 33^(@))-(cot 70^(@))/(tan 20^(@))-sqrt(2)cos 45^(@)`

Answer

Let's evaluate the expression step by step:

Given expression:
\[ 2 \left( \tan 57^\circ \right) / \left( \cot 33^\circ \right) - \left( \cot 70^\circ \right) / \left( \tan 20^\circ \right) - \sqrt{2} \cos 45^\circ \]

### Step 1: Rewrite the trigonometric functions using complementary angles
We know that:
\[ \cot(90^\circ - \theta) = \tan(\theta) \]

Using this property:
- For \( \tan 57^\circ \), we can write:
\[ \tan 57^\circ = \cot(90^\circ - 57^\circ) = \cot 33^\circ \]
- For \( \tan 20^\circ \), we can write:
\[ \tan 20^\circ = \cot(90^\circ - 20^\circ) = \cot 70^\circ \]

Now, substituting these into the expression:
\[ 2 \left( \cot 33^\circ \right) / \left( \cot 33^\circ \right) - \left( \cot 70^\circ \right) / \left( \cot 70^\circ \right) - \sqrt{2} \cos 45^\circ \]

### Step 2: Simplify the expression
Now, simplifying the fractions:
\[ 2 \cdot 1 - 1 - \sqrt{2} \cos 45^\circ \]

### Step 3: Evaluate \( \cos 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

Substituting this value into the expression:
\[ 2 - 1 - \sqrt{2} \cdot \frac{1}{\sqrt{2}} \]

### Step 4: Simplify further
The term \( \sqrt{2} \cdot \frac{1}{\sqrt{2}} \) simplifies to 1:
\[ 2 - 1 - 1 \]

### Step 5: Final calculation
Now, calculate:
\[ 2 - 1 - 1 = 0 \]

Thus, the final answer is:
\[ \boxed{0} \]

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