If `A=(41pi)/(12)`, then what is the value of `(1-3tan^(2)A)/(3tanA-tan^(3)A)`?

Options

A. `-1`
B. 1
C. `1//3`
D. 3

Explanation

To solve the problem, we need to find the value of

\[ \frac{1 - 3\tan^2 A}{3\tan A - \tan^3 A} \]

given that \( A = \frac{41\pi}{12} \).

### Step 1: Use the identity for \(\tan 3A\)

We know from trigonometric identities that:

\[ \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \]

This means we can express our original expression as:

\[ \frac{1 - 3\tan^2 A}{3\tan A - \tan^3 A} = \frac{1}{\tan 3A} \]

### Step 2: Calculate \(3A\)

Now, we need to calculate \(3A\):

\[ 3A = 3 \times \frac{41\pi}{12} = \frac{123\pi}{12} = \frac{41\pi}{4} \]

### Step 3: Find \(\tan 3A\)

Next, we need to find \(\tan\left(\frac{41\pi}{4}\right)\). To do this, we can reduce the angle:

\[ \frac{41\pi}{4} = 10\pi + \frac{\pi}{4} \]

Since \(10\pi\) is an even multiple of \(\pi\), it does not affect the tangent function:

\[ \tan\left(\frac{41\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 4: Substitute back into the expression

Now substituting back into our expression:

\[ \frac{1}{\tan 3A} = \frac{1}{1} = 1 \]

### Final Answer

Thus, the value of

\[ \frac{1 - 3\tan^2 A}{3\tan A - \tan^3 A} = 1 \]

### Conclusion

The answer is \(1\), which corresponds to option B.

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