The value of `(cot 40^(@))/(tan50^(@))-(1)/(2)(cos35^(@))/(sin55^(@))` is :

Options

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

Answer

To solve the expression \(\frac{\cot 40^\circ}{\tan 50^\circ} - \frac{1}{2} \frac{\cos 35^\circ}{\sin 55^\circ}\), we will use trigonometric identities and properties.

### Step 1: Simplify \(\frac{\cot 40^\circ}{\tan 50^\circ}\)

Recall that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can rewrite \(\cot 40^\circ\) as:
\[ \cot 40^\circ = \frac{1}{\tan 40^\circ} \]
Now, substituting this into our expression gives:
\[ \frac{\cot 40^\circ}{\tan 50^\circ} = \frac{1}{\tan 40^\circ \tan 50^\circ} \]

### Step 2: Use the identity \(\tan(90^\circ - \theta) = \cot \theta\)

We know that:
\[ \tan(90^\circ - 50^\circ) = \cot 50^\circ \]
Thus, we can express \(\tan 50^\circ\) as:
\[ \tan 50^\circ = \cot 40^\circ \]
This means:
\[ \frac{\cot 40^\circ}{\tan 50^\circ} = \frac{\cot 40^\circ}{\cot 40^\circ} = 1 \]

### Step 3: Simplify \(\frac{1}{2} \frac{\cos 35^\circ}{\sin 55^\circ}\)

Using the identity \(\sin(90^\circ - \theta) = \cos \theta\):
\[ \sin 55^\circ = \cos(90^\circ - 55^\circ) = \cos 35^\circ \]
Thus, we can rewrite \(\frac{1}{2} \frac{\cos 35^\circ}{\sin 55^\circ}\) as:
\[ \frac{1}{2} \frac{\cos 35^\circ}{\cos 35^\circ} = \frac{1}{2} \]

### Step 4: Combine the results

Now we can substitute back into our original expression:
\[ 1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2} \]

### Final Answer
The value of the expression is:
\[ \frac{1}{2} \]