Evaluate :
`(2sin68^(@))/(cos22^(@))-(2cot15^(@))/(5 tan 75^(@))-(3tan45^(@)tan20^(@)tan40^(@)tan50^(@)tan70^(@))/(5)`

Options

A. 1
B. 4
C. 2
D. 3

Answer

To evaluate the expression

\[ E = \frac{2 \sin 68^\circ}{\cos 22^\circ} - \frac{2 \cot 15^\circ}{5 \tan 75^\circ} - \frac{3 \tan 45^\circ \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ}{5} \]

let's simplify it step by step.

### Step 1: Simplify \( \frac{2 \sin 68^\circ}{\cos 22^\circ} \)

Using the identity \( \cos(90^\circ - \theta) = \sin \theta \), we have:

\[ \cos 22^\circ = \sin(90^\circ - 22^\circ) = \sin 68^\circ \]

Thus,

\[ \frac{2 \sin 68^\circ}{\cos 22^\circ} = \frac{2 \sin 68^\circ}{\sin 68^\circ} = 2 \]

### Step 2: Simplify \( \frac{2 \cot 15^\circ}{5 \tan 75^\circ} \)

Using the identity \( \tan(90^\circ - \theta) = \cot \theta \):

\[ \tan 75^\circ = \cot(90^\circ - 75^\circ) = \cot 15^\circ \]

Therefore,

\[ \frac{2 \cot 15^\circ}{5 \tan 75^\circ} = \frac{2 \cot 15^\circ}{5 \cot 15^\circ} = \frac{2}{5} \]

### Step 3: Simplify \( \frac{3 \tan 45^\circ \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ}{5} \)

We know that \( \tan 45^\circ = 1 \). Therefore, we can rewrite the expression as:

\[ \frac{3 \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ}{5} \]

Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we have:

\[ \tan 70^\circ = \cot 20^\circ \quad \text{and} \quad \tan 50^\circ = \cot 40^\circ \]

Thus,

\[ \tan 20^\circ \tan 70^\circ = 1 \quad \text{and} \quad \tan 40^\circ \tan 50^\circ = 1 \]

So,

\[ \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ = 1 \]

This means:

\[ \frac{3 \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ}{5} = \frac{3 \cdot 1}{5} = \frac{3}{5} \]

### Step 4: Combine all parts

Now we can combine all the simplified parts:

\[ E = 2 - \frac{2}{5} - \frac{3}{5} \]

First, combine the fractions:

\[ E = 2 - \frac{2 + 3}{5} = 2 - \frac{5}{5} = 2 - 1 = 1 \]

### Final Answer

Thus, the value of the expression is

\[ \boxed{1} \]