If x = 4 cos A + 5 sin A and y = 4 sin A - 5 cos A, then the value of `x^(2) + y^(2)` is:

Options

A. 25
B. 0
C. 16
D. 41

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 \) given the equations:

\[ x = 4 \cos A + 5 \sin A \]
\[ y = 4 \sin A - 5 \cos A \]

### Step 1: Calculate \( x^2 \) and \( y^2 \)

First, we calculate \( x^2 \) and \( y^2 \):

\[ x^2 = (4 \cos A + 5 \sin A)^2 = (4 \cos A)^2 + 2(4 \cos A)(5 \sin A) + (5 \sin A)^2 \]
\[ = 16 \cos^2 A + 40 \cos A \sin A + 25 \sin^2 A \]

\[ y^2 = (4 \sin A - 5 \cos A)^2 = (4 \sin A)^2 - 2(4 \sin A)(5 \cos A) + (5 \cos A)^2 \]
\[ = 16 \sin^2 A - 40 \sin A \cos A + 25 \cos^2 A \]

### Step 2: Combine \( x^2 \) and \( y^2 \)

Now, we add \( x^2 \) and \( y^2 \):

\[ x^2 + y^2 = (16 \cos^2 A + 40 \cos A \sin A + 25 \sin^2 A) + (16 \sin^2 A - 40 \sin A \cos A + 25 \cos^2 A) \]

### Step 3: Simplify the expression

Combining like terms:

\[ = (16 \cos^2 A + 25 \cos^2 A) + (16 \sin^2 A + 25 \sin^2 A) + (40 \cos A \sin A - 40 \sin A \cos A) \]
\[ = 41 \cos^2 A + 41 \sin^2 A \]

### Step 4: Apply the Pythagorean identity

Using the identity \( \cos^2 A + \sin^2 A = 1 \):

\[ x^2 + y^2 = 41(\cos^2 A + \sin^2 A) = 41 \cdot 1 = 41 \]

### Final Answer

Thus, the value of \( x^2 + y^2 \) is:

\[ \boxed{41} \]