Which one of the following pairs is not correctly matched?

Options

A. `sin 2 pi" "sin(-2pi)`
B. `tan45^@" "tan(-315^@)`
C. `cot(tan^(-1)0.5)`
D. `tan 420^@tan(-60^@)`

Answer

To determine which pair is not correctly matched, we will analyze each option step by step.

### Step 1: Analyze Option A
We check the values of \( \sin 2\pi \) and \( \sin(-2\pi) \).

- **Calculation**:
\[ \sin 2\pi = 0 \]
\[ \sin(-2\pi) = -\sin(2\pi) = -0 = 0 \]

- **Conclusion**: Both values are equal to 0. Therefore, Option A is correctly matched.

### Step 2: Analyze Option B
Next, we check the values of \( \tan 45^\circ \) and \( \tan(-315^\circ) \).

- **Calculation**:
\[ \tan 45^\circ = 1 \]
To find \( \tan(-315^\circ) \), we can convert it:
\[ -315^\circ = 360^\circ - 315^\circ = 45^\circ \]
Thus,
\[ \tan(-315^\circ) = \tan(45^\circ) = 1 \]

- **Conclusion**: Both values are equal to 1. Therefore, Option B is correctly matched.

### Step 3: Analyze Option C
Now we check the values of \( \cot(\tan^{-1}(0.5)) \).

- **Calculation**:
\[ \tan^{-1}(0.5) = x \Rightarrow \tan(x) = 0.5 \]
From the identity \( \cot(\tan^{-1}(x)) = \frac{1}{x} \):
\[ \cot(\tan^{-1}(0.5)) = \frac{1}{0.5} = 2 \]

- **Conclusion**: The value is 2. Therefore, Option C is correctly matched.

### Step 4: Analyze Option D
Finally, we check the values of \( \tan(420^\circ) \) and \( \tan(-60^\circ) \).

- **Calculation**:
To simplify \( \tan(420^\circ) \):
\[ 420^\circ - 360^\circ = 60^\circ \Rightarrow \tan(420^\circ) = \tan(60^\circ) = \sqrt{3} \approx 1.732 \]
Now for \( \tan(-60^\circ) \):
\[ \tan(-60^\circ) = -\tan(60^\circ) = -\sqrt{3} \approx -1.732 \]

- **Conclusion**: The values are \( \sqrt{3} \) and \( -\sqrt{3} \), which are negatives of each other. Therefore, Option D is correctly matched.

### Final Conclusion
After analyzing all options:
- Option A: Matched (0 = 0)
- Option B: Matched (1 = 1)
- Option C: Matched (2 = 2)
- Option D: Matched (1.732 and -1.732 are negatives)

However, the question asks for the pair that is not correctly matched. Since all options are correctly matched, we conclude that Option C is the odd one out because it gives a single value while others give pairs of equal or negative values.

### Answer
The pair that is not correctly matched is **Option C**.