The value of `cos(tan^-1 (tan 2))` is

Options

A. `1//sqrt(5)`
B. `-1//sqrt(5)`
C. cos 2
D. `-cos 2`

Answer

To solve the problem \( \cos(\tan^{-1}(\tan 2)) \), we will follow these steps:

### Step 1: Identify the angle
Let \( x = 2 \). We need to evaluate \( \cos(\tan^{-1}(\tan x)) \).

### Step 2: Determine the range of \( x \)
We know that \( 2 \) is between \( \frac{\pi}{2} \) and \( \pi \). To clarify:
- \( \frac{\pi}{2} \approx 1.57 \)
- \( \pi \approx 3.14 \)

Since \( 2 \) is greater than \( \frac{\pi}{2} \) and less than \( \pi \), we can conclude:
\[ \frac{\pi}{2} < 2 < \pi \]

### Step 3: Apply the property of inverse tangent
For angles \( x \) in the range \( \frac{\pi}{2} < x < \pi \), we have the property:
\[ \tan^{-1}(\tan x) = x - \pi \]
Thus, we can write:
\[ \tan^{-1}(\tan 2) = 2 - \pi \]

### Step 4: Substitute back into the cosine function
Now we substitute this result into the cosine function:
\[ \cos(\tan^{-1}(\tan 2)) = \cos(2 - \pi) \]

### Step 5: Simplify using cosine properties
Using the cosine subtraction formula, we know:
\[ \cos(2 - \pi) = -\cos(2) \]
This is because \( \cos(\theta - \pi) = -\cos(\theta) \).

### Step 6: Final result
Thus, we find:
\[ \cos(\tan^{-1}(\tan 2)) = -\cos(2) \]

### Conclusion
The value of \( \cos(\tan^{-1}(\tan 2)) \) is \( -\cos(2) \).

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