Without using trigonometric tables, find the value of : `(2)/(3) ((sec 56^(@))/( cosec 34^(@))) - 2 cos^(2) 20^(@) + (1)/(2) cot 28^(@) cot 35^(@) cot 45^(@) cot 62^(@) cot 55^(@) - 2 cos ^(2) 70^(@)`

Options

A. `(4)/(5)`
B. `-(3)/(4)`
C. `-(5)/(6)`
D. 1

Answer

To solve the expression

\[ \frac{2}{3} \left( \frac{\sec 56^\circ}{\csc 34^\circ} \right) - 2 \cos^2 20^\circ + \frac{1}{2} \cot 28^\circ \cot 35^\circ \cot 45^\circ \cot 62^\circ \cot 55^\circ - 2 \cos^2 70^\circ, \]

we will break it down step by step.

### Step 1: Simplify \(\frac{\sec 56^\circ}{\csc 34^\circ}\)

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

\[ \sec 56^\circ = \csc(90^\circ - 56^\circ) = \csc 34^\circ. \]

Thus,

\[ \frac{\sec 56^\circ}{\csc 34^\circ} = 1. \]

### Step 2: Substitute into the expression

Now substituting this back into the expression, we get:

\[ \frac{2}{3} \cdot 1 - 2 \cos^2 20^\circ + \frac{1}{2} \cot 28^\circ \cot 35^\circ \cot 45^\circ \cot 62^\circ \cot 55^\circ - 2 \cos^2 70^\circ. \]

### Step 3: Simplify \(-2 \cos^2 20^\circ - 2 \cos^2 70^\circ\)

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

\[ \cos^2 70^\circ = \sin^2 20^\circ. \]

Thus,

\[ -2 \cos^2 20^\circ - 2 \cos^2 70^\circ = -2 \cos^2 20^\circ - 2 \sin^2 20^\circ = -2 (\cos^2 20^\circ + \sin^2 20^\circ). \]

Using the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\), we find:

\[ -2 (\cos^2 20^\circ + \sin^2 20^\circ) = -2 \cdot 1 = -2. \]

### Step 4: Simplify \(\frac{1}{2} \cot 28^\circ \cot 35^\circ \cot 45^\circ \cot 62^\circ \cot 55^\circ\)

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

\[ \cot 28^\circ = \tan 62^\circ, \quad \cot 35^\circ = \tan 55^\circ. \]

Thus, we can pair them:

\[ \cot 28^\circ \cot 62^\circ = 1 \quad \text{and} \quad \cot 35^\circ \cot 55^\circ = 1. \]

So,

\[ \cot 28^\circ \cot 35^\circ \cot 45^\circ \cot 62^\circ \cot 55^\circ = 1 \cdot 1 \cdot 1 = 1. \]

Thus,

\[ \frac{1}{2} \cot 28^\circ \cot 35^\circ \cot 45^\circ \cot 62^\circ \cot 55^\circ = \frac{1}{2}. \]

### Step 5: Combine all parts

Now we can combine all parts:

\[ \frac{2}{3} - 2 + \frac{1}{2}. \]

To combine these, we need a common denominator. The least common multiple of 3 and 2 is 6.

Converting each term:

\[ \frac{2}{3} = \frac{4}{6}, \quad -2 = -\frac{12}{6}, \quad \frac{1}{2} = \frac{3}{6}. \]

Now substituting back, we have:

\[ \frac{4}{6} - \frac{12}{6} + \frac{3}{6} = \frac{4 - 12 + 3}{6} = \frac{-5}{6}. \]

### Final Answer

The final value of the expression is:

\[ \boxed{-\frac{5}{6}}. \]