The value of `tan^(-1)((sin2-1)/(cos2))` is

Options

A. `(pi)/(2)-1`
B. `1-(pi)/(4)`
C. `2-(pi)/(2)`
D. `(pi)/(4)-1`

Explanation

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

### Step 1: Rewrite \(\sin 2\) and \(\cos 2\)
Using the double angle formulas:
\[ \sin 2 = 2 \sin 1 \cos 1 \]
\[ \cos 2 = \cos^2 1 - \sin^2 1 \]
So we can substitute these into our expression.

### Step 2: Substitute into the expression
Now substituting these into the original expression:
\[ \tan^{-1}\left(\frac{2 \sin 1 \cos 1 - 1}{\cos^2 1 - \sin^2 1}\right) \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ 2 \sin 1 \cos 1 - 1 \]
We can rewrite \(1\) as \(\sin^2 1 + \cos^2 1\):
\[ 2 \sin 1 \cos 1 - (\sin^2 1 + \cos^2 1) = 2 \sin 1 \cos 1 - \sin^2 1 - \cos^2 1 \]

### Step 4: Factor the numerator
The numerator can be factored as:
\[ -(\sin^2 1 + \cos^2 1 - 2 \sin 1 \cos 1) = -(\sin 1 - \cos 1)^2 \]

### Step 5: Substitute back into the expression
Now we have:
\[ \tan^{-1}\left(\frac{-(\sin 1 - \cos 1)^2}{\cos^2 1 - \sin^2 1}\right) \]

### Step 6: Simplify the denominator
The denominator can be rewritten as:
\[ \cos^2 1 - \sin^2 1 = (\cos 1 + \sin 1)(\cos 1 - \sin 1) \]

### Step 7: Final expression
Thus, we can rewrite our expression as:
\[ \tan^{-1}\left(-\frac{(\sin 1 - \cos 1)^2}{(\cos 1 + \sin 1)(\cos 1 - \sin 1)}\right) \]

### Step 8: Recognize the tangent identity
Using the identity for tangent subtraction:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
we can express our result in terms of angles.

### Step 9: Final calculation
Recognizing that:
\[ \tan^{-1}\left(-\frac{(\sin 1 - \cos 1)^2}{(\cos 1 + \sin 1)(\cos 1 - \sin 1)}\right) = -\left(\frac{\pi}{4} - 1\right) \]
Thus, the final answer is:
\[ 1 - \frac{\pi}{4} \]

### Summary
The value of \( \tan^{-1}\left(\frac{\sin 2 - 1}{\cos 2}\right) \) is:
\[ 1 - \frac{\pi}{4} \]