If `tantheta-cottheta=a and costheta+sintheta=b`, then `(b^(2)-1)^(2)(a^(2)+4)=?`

Options

A. 2
B. `-4`
C. `pm4`
D. 4

Answer

To solve the problem step by step, we will use the given equations and substitute values for \( \theta \).

### Step 1: Set the equations
We are given:
1. \( \tan \theta - \cot \theta = a \)
2. \( \cos \theta + \sin \theta = b \)

### Step 2: Choose a value for \( \theta \)
To simplify our calculations, we will choose \( \theta = 45^\circ \). This is a common angle that makes calculations easier.

### Step 3: Calculate \( a \)
Using \( \theta = 45^\circ \):
- \( \tan 45^\circ = 1 \)
- \( \cot 45^\circ = 1 \)

Now substituting these values into the first equation:
\[ a = \tan 45^\circ - \cot 45^\circ = 1 - 1 = 0 \]

### Step 4: Calculate \( b \)
Now, we will calculate \( b \) using the second equation:
\[ b = \cos 45^\circ + \sin 45^\circ \]
Since:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)

Substituting these values:
\[ b = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 5: Calculate \( b^2 - 1 \)
Now we need to calculate \( b^2 - 1 \):
\[ b^2 = (\sqrt{2})^2 = 2 \]
So,
\[ b^2 - 1 = 2 - 1 = 1 \]

### Step 6: Calculate \( (b^2 - 1)^2 (a^2 + 4) \)
Now we can calculate the expression:
\[ (b^2 - 1)^2 = 1^2 = 1 \]
And,
\[ a^2 + 4 = 0^2 + 4 = 4 \]

Now, substituting these values into the expression:
\[ (b^2 - 1)^2 (a^2 + 4) = 1 \cdot 4 = 4 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{4} \]