Evaluate the following expression:
`sec(tan{tan^(-1)(-(pi)/3)})`

Explanation

To evaluate the expression \( \sec(\tan(\tan^{-1}(-\frac{\pi}{3}))) \), we can follow these steps:

### Step 1: Simplify the Inner Function
We start with the innermost function, which is \( \tan^{-1}(-\frac{\pi}{3}) \). The tangent inverse function gives us an angle whose tangent is the input value.

### Step 2: Determine the Angle
The value \( -\frac{\pi}{3} \) corresponds to an angle in the fourth quadrant. Therefore, we can express this as:
\[ \tan^{-1}(-\frac{\pi}{3}) = -\frac{\pi}{3} \]

### Step 3: Substitute Back into the Expression
Now we substitute this back into the expression:
\[ \tan(-\frac{\pi}{3}) \]

### Step 4: Calculate the Tangent
The tangent of \( -\frac{\pi}{3} \) can be calculated as:
\[ \tan(-\frac{\pi}{3}) = -\tan(\frac{\pi}{3}) = -\sqrt{3} \]

### Step 5: Substitute into the Secant Function
Now we substitute this value into the secant function:
\[ \sec(-\sqrt{3}) \]

### Step 6: Evaluate the Secant
The secant function is defined as the reciprocal of the cosine function. We need to find the angle whose tangent is \( -\sqrt{3} \). The angle corresponding to \( -\sqrt{3} \) is \( -\frac{\pi}{3} \), which means:
\[ \sec(-\frac{\pi}{3}) = \frac{1}{\cos(-\frac{\pi}{3})} \]

### Step 7: Calculate the Cosine
The cosine of \( -\frac{\pi}{3} \) is:
\[ \cos(-\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2} \]

### Step 8: Final Calculation
Thus, we have:
\[ \sec(-\frac{\pi}{3}) = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
The value of the expression \( \sec(\tan(\tan^{-1}(-\frac{\pi}{3}))) \) is:
\[ \boxed{2} \]

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