`cos[tan^(-1){tan((15pi)/4)}]`

Options

A. `(1)/sqrt(2)`
B. `-(1)/sqrt(2)`
C. 1
D. none of these

Explanation

To solve the expression \( \cos(\tan^{-1}(\tan(\frac{15\pi}{4}))) \), we can follow these steps:

### Step 1: Simplify \( \tan(\frac{15\pi}{4}) \)
The angle \( \frac{15\pi}{4} \) can be simplified by subtracting \( 2\pi \) (which is equivalent to \( \frac{8\pi}{4} \)):
\[ \frac{15\pi}{4} - 2\pi = \frac{15\pi}{4} - \frac{8\pi}{4} = \frac{7\pi}{4} \]
Thus, we have:
\[ \tan\left(\frac{15\pi}{4}\right) = \tan\left(\frac{7\pi}{4}\right) \]

### Step 2: Evaluate \( \tan\left(\frac{7\pi}{4}\right) \)
The angle \( \frac{7\pi}{4} \) is in the fourth quadrant where the tangent function is negative. The reference angle is \( \frac{\pi}{4} \), so:
\[ \tan\left(\frac{7\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1 \]

### Step 3: Substitute into the expression
Now we substitute back into the expression:
\[ \cos(\tan^{-1}(-1)) \]

### Step 4: Evaluate \( \tan^{-1}(-1) \)
The angle whose tangent is \(-1\) is \( -\frac{\pi}{4} \) (since the range of \( \tan^{-1} \) is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)):
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

### Step 5: Calculate \( \cos(-\frac{\pi}{4}) \)
Using the property of cosine that \( \cos(-\theta) = \cos(\theta) \):
\[ \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) \]
The value of \( \cos(\frac{\pi}{4}) \) is:
\[ \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the value of \( \cos(\tan^{-1}(\tan(\frac{15\pi}{4}))) \) is:
\[ \frac{1}{\sqrt{2}} \]