The value of `(cot54^(@))/(tan36^(@))+(tan20^(@))/(cot70^(@))` is

Options

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

Answer

To solve the expression \(\frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ}\), we can simplify it step by step.

### Step 1: Rewrite the cotangent and tangent functions
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Using this, we can rewrite \(\cot 54^\circ\) and \(\cot 70^\circ\):
\[ \cot 54^\circ = \frac{1}{\tan 54^\circ} \]
\[ \cot 70^\circ = \frac{1}{\tan 70^\circ} \]

### Step 2: Substitute and simplify the expression
Now, substituting these values into the original expression:
\[ \frac{\cot 54^\circ}{\tan 36^\circ} = \frac{1/\tan 54^\circ}{\tan 36^\circ} = \frac{1}{\tan 54^\circ \tan 36^\circ} \]
And for the second part:
\[ \frac{\tan 20^\circ}{\cot 70^\circ} = \tan 20^\circ \cdot \tan 70^\circ \]
Thus, the expression becomes:
\[ \frac{1}{\tan 54^\circ \tan 36^\circ} + \tan 20^\circ \tan 70^\circ \]

### Step 3: Use the complementary angle identities
Using the identity \(\tan(90^\circ - \theta) = \cot \theta\), we can find that:
\[ \tan 70^\circ = \cot 20^\circ \]
So, we can rewrite \(\tan 20^\circ \tan 70^\circ\) as:
\[ \tan 20^\circ \cdot \cot 20^\circ = 1 \]

### Step 4: Substitute back into the expression
Now substituting this back into our expression:
\[ \frac{1}{\tan 54^\circ \tan 36^\circ} + 1 \]

### Step 5: Simplify \(\tan 54^\circ\) and \(\tan 36^\circ\)
Using the identity \(\tan(90^\circ - \theta) = \cot \theta\), we know:
\[ \tan 54^\circ = \cot 36^\circ \]
Thus:
\[ \tan 54^\circ \tan 36^\circ = \cot 36^\circ \tan 36^\circ = 1 \]

### Step 6: Final substitution
Substituting this back into our expression:
\[ \frac{1}{1} + 1 = 1 + 1 = 2 \]

### Final Answer
The value of the expression is:
\[ \boxed{2} \]