The value of ` (cot 54^(@))/(tan 36^(@)) + (tan 20^(@))/(cot 70^(@))` is

Options

A. 0
B. 2
C. 3
D. 1

Explanation

To solve the expression \( \frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ} \), we will use trigonometric identities and properties.

### Step-by-Step Solution:

1. **Rewrite cotangent and tangent functions**:
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
and also,
\[ \cot(90^\circ - \theta) = \tan \theta \]
Therefore, we can rewrite:
\[ \cot 54^\circ = \tan(90^\circ - 54^\circ) = \tan 36^\circ \]
and
\[ \cot 70^\circ = \tan(90^\circ - 70^\circ) = \tan 20^\circ \]

2. **Substituting into the expression**:
Now substituting these values into the original expression:
\[ \frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ} = \frac{\tan 36^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\tan 20^\circ} \]

3. **Simplifying the fractions**:
Each fraction simplifies to:
\[ \frac{\tan 36^\circ}{\tan 36^\circ} = 1 \]
and
\[ \frac{\tan 20^\circ}{\tan 20^\circ} = 1 \]
Therefore, the expression becomes:
\[ 1 + 1 \]

4. **Final Calculation**:
Adding these results gives:
\[ 1 + 1 = 2 \]

Thus, the value of the expression \( \frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ} \) is \( \boxed{2} \).