Evaluate:
` (cos 58^@)/(sin 32^@)+(sin 22^@)/(cos 68^@)-(cos 38^@cosec 52^@)/(tan 18^@tan 35 ^@ tan 60^@tan 72^@tan 55^@)`.

Options

A. `(1)/(3)(6-sqrt(3))`
B. `(2)/(3)(6-sqrt(3))`
C. `(1)/(3)(3-sqrt(6))`
D. `(2)/(3)(3-sqrt(6))`

Explanation

To evaluate the expression

\[ \frac{\cos 58^\circ}{\sin 32^\circ} + \frac{\sin 22^\circ}{\cos 68^\circ} - \frac{\cos 38^\circ \cdot \csc 52^\circ}{\tan 18^\circ \tan 35^\circ \tan 60^\circ \tan 72^\circ \tan 55^\circ}, \]

we will simplify each term step by step.

### Step 1: Simplifying the first term

We know that

\[ \cos 58^\circ = \sin(90^\circ - 58^\circ) = \sin 32^\circ. \]

Thus, we can rewrite the first term:

\[ \frac{\cos 58^\circ}{\sin 32^\circ} = \frac{\sin 32^\circ}{\sin 32^\circ} = 1. \]

### Step 2: Simplifying the second term

Next, we simplify the second term:

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

So, we can rewrite the second term:

\[ \frac{\sin 22^\circ}{\cos 68^\circ} = \frac{\sin 22^\circ}{\sin 22^\circ} = 1. \]

### Step 3: Simplifying the third term

Now, we simplify the third term:

\[ \csc 52^\circ = \frac{1}{\sin 52^\circ}. \]

Also, we know that

\[ \cos 38^\circ = \sin(90^\circ - 38^\circ) = \sin 52^\circ. \]

Thus, we can rewrite the third term:

\[ \frac{\cos 38^\circ \cdot \csc 52^\circ}{\tan 18^\circ \tan 35^\circ \tan 60^\circ \tan 72^\circ \tan 55^\circ} = \frac{\sin 52^\circ \cdot \frac{1}{\sin 52^\circ}}{\tan 18^\circ \tan 35^\circ \tan 60^\circ \tan 72^\circ \tan 55^\circ} = \frac{1}{\tan 18^\circ \tan 35^\circ \tan 60^\circ \tan 72^\circ \tan 55^\circ}. \]

### Step 4: Evaluating the product of tangents

The product of tangents can be simplified using the identity:

\[ \tan(90^\circ - x) = \cot x. \]

Thus, we have:

\[ \tan 72^\circ = \cot 18^\circ, \quad \tan 60^\circ = \sqrt{3}, \quad \tan 55^\circ = \cot 35^\circ. \]

So, the product becomes:

\[ \tan 18^\circ \tan 35^\circ \tan 60^\circ \tan 72^\circ \tan 55^\circ = \tan 18^\circ \tan 35^\circ \cdot \sqrt{3} \cdot \cot 18^\circ \cdot \cot 35^\circ = \sqrt{3}. \]

### Step 5: Final expression

Now we can combine everything:

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

To express this in a single fraction, we can write:

\[ 2 = \frac{2\sqrt{3}}{\sqrt{3}}. \]

Thus, the final expression becomes:

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

### Final Answer

The evaluated expression is:

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

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