Evaluate :
`(cot40^(@))/(tan50^(@))-(1)/(2)((cos30^(@))/(sin60^(@)))`

Options

A. `(3)/(2)`
B. 1
C. `(1)/(2)`
D. 0

Answer

To evaluate the expression

\[ \frac{\cot 40^\circ}{\tan 50^\circ} - \frac{1}{2} \left( \frac{\cos 30^\circ}{\sin 60^\circ} \right), \]

we will follow these steps:

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

We know that:

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

Thus,

\[ \tan 50^\circ = \cot(90^\circ - 50^\circ) = \cot 40^\circ. \]

This means:

\[ \frac{\cot 40^\circ}{\tan 50^\circ} = \frac{\cot 40^\circ}{\cot 40^\circ} = 1. \]

### Step 2: Simplify \(\frac{1}{2} \left( \frac{\cos 30^\circ}{\sin 60^\circ} \right)\)

We know the values of \(\cos 30^\circ\) and \(\sin 60^\circ\):

\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}. \]

Thus,

\[ \frac{\cos 30^\circ}{\sin 60^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}} = 1. \]

Now, substituting this back into our expression gives:

\[ \frac{1}{2} \left( \frac{\cos 30^\circ}{\sin 60^\circ} \right) = \frac{1}{2} \times 1 = \frac{1}{2}. \]

### Step 3: 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 Result

Thus, the value of the expression is

\[ \frac{1}{2}. \]