A and B are complementary angles, then find the value of `sinAcosB+cosAsinB+2tanAtanB-sec^(2)A+cot^(2)B`.

Options

A. 5
B. `sqrt(2)`
C. 2
D. 3

Answer

To solve the expression \( \sin A \cos B + \cos A \sin B + 2 \tan A \tan B - \sec^2 A + \cot^2 B \) given that \( A \) and \( B \) are complementary angles, we can follow these steps:

### Step 1: Understand the relationship between complementary angles
Since \( A \) and \( B \) are complementary angles, we have:
\[ A + B = 90^\circ \quad \text{or} \quad B = 90^\circ - A \]

### Step 2: Substitute \( B \) in the expression
Using the relationship \( B = 90^\circ - A \), we can rewrite the expression:
\[ \sin A \cos(90^\circ - A) + \cos A \sin(90^\circ - A) + 2 \tan A \tan(90^\circ - A) - \sec^2 A + \cot^2(90^\circ - A) \]

### Step 3: Simplify using trigonometric identities
Using the trigonometric identities:
- \( \cos(90^\circ - A) = \sin A \)
- \( \sin(90^\circ - A) = \cos A \)
- \( \tan(90^\circ - A) = \cot A \)
- \( \cot(90^\circ - A) = \tan A \)

Substituting these identities into the expression gives:
\[ \sin A \sin A + \cos A \cos A + 2 \tan A \cot A - \sec^2 A + \tan^2 A \]

### Step 4: Further simplify the expression
This can be simplified to:
\[ \sin^2 A + \cos^2 A + 2 \cdot 1 - \sec^2 A + \tan^2 A \]
Using the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \):
\[ 1 + 2 - \sec^2 A + \tan^2 A \]

### Step 5: Use the identity \( \sec^2 A = 1 + \tan^2 A \)
Substituting \( \sec^2 A \) gives:
\[ 1 + 2 - (1 + \tan^2 A) + \tan^2 A \]
This simplifies to:
\[ 1 + 2 - 1 - \tan^2 A + \tan^2 A = 2 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2} \]