Without using the trigonometric tables, evaluate the following :
`(11)/(7) (sin 70^(@))/(cos 20^(@)) - (4)/(7) (cos 53^(@) "cosec" 37^(@))/(tan 15^(@) tan 35^(@) tan 55^(@) tan 75^(@))`

Answer

To evaluate the expression

\[ \frac{11}{7} \frac{\sin 70^\circ}{\cos 20^\circ} - \frac{4}{7} \frac{\cos 53^\circ \csc 37^\circ}{\tan 15^\circ \tan 35^\circ \tan 55^\circ \tan 75^\circ} \]

we can simplify it step by step using trigonometric identities.

### Step 1: Simplifying the first term
We know that:

\[ \sin 70^\circ = \cos 20^\circ \]

This is because \(70^\circ + 20^\circ = 90^\circ\), and we can use the identity \(\sin(90^\circ - \theta) = \cos(\theta)\). Therefore, we can rewrite the first term:

\[ \frac{11}{7} \frac{\sin 70^\circ}{\cos 20^\circ} = \frac{11}{7} \frac{\cos 20^\circ}{\cos 20^\circ} = \frac{11}{7} \cdot 1 = \frac{11}{7} \]

### Step 2: Simplifying the second term
Next, we simplify the second term:

\[ \frac{4}{7} \frac{\cos 53^\circ \csc 37^\circ}{\tan 15^\circ \tan 35^\circ \tan 55^\circ \tan 75^\circ} \]

First, recall that \(\csc 37^\circ = \frac{1}{\sin 37^\circ}\) and \(\cos 53^\circ = \sin 37^\circ\) (since \(53^\circ + 37^\circ = 90^\circ\)). Thus, we can rewrite:

\[ \cos 53^\circ \csc 37^\circ = \sin 37^\circ \cdot \frac{1}{\sin 37^\circ} = 1 \]

So the second term simplifies to:

\[ \frac{4}{7} \cdot \frac{1}{\tan 15^\circ \tan 35^\circ \tan 55^\circ \tan 75^\circ} \]

### Step 3: Simplifying the product of tangents
Now, we need to simplify the product of tangents:

\[ \tan 15^\circ \tan 75^\circ = 1 \quad \text{(since } 15^\circ + 75^\circ = 90^\circ\text{)} \]

\[ \tan 35^\circ \tan 55^\circ = 1 \quad \text{(since } 35^\circ + 55^\circ = 90^\circ\text{)} \]

Thus, we have:

\[ \tan 15^\circ \tan 35^\circ \tan 55^\circ \tan 75^\circ = 1 \cdot 1 = 1 \]

### Step 4: Putting it all together
Now we can substitute back into the expression:

\[ \frac{4}{7} \cdot \frac{1}{1} = \frac{4}{7} \]

### Step 5: Final calculation
Now we combine the simplified terms:

\[ \frac{11}{7} - \frac{4}{7} = \frac{11 - 4}{7} = \frac{7}{7} = 1 \]

Thus, the final value of the expression is:

\[ \boxed{1} \]