Suppose that `acostheta = b` and `csintheta = d`, for `theta` and some constants a, b, c, d. Then which of the following is true?

Options

A. `a^(2)c^(2) = b^(2)c^(2) + a^(2)d^(2)`
B. `a^(2)d^(2) = b^(2)c^(2) + a^(2)c^(2)`
C. `b^(2)c^(2) =a^(2)d^(2) + a^(2)c^(2)`
D. `a^(2)b^(2) = b^(2)c^(2) + c^(2)d^(2)`

Answer

To solve the problem, we start with the given equations:

1. \( a \cos \theta = b \)
2. \( c \sin \theta = d \)

We need to explore the relationships between these equations and find which of the provided options is true.

### Step 1: Square the first equation
From the first equation, square both sides:
\[ (a \cos \theta)^2 = b^2 \]
This simplifies to:
\[ a^2 \cos^2 \theta = b^2 \tag{1} \]

### Step 2: Multiply by \( c^2 \)
Now, multiply both sides of equation (1) by \( c^2 \):
\[ a^2 c^2 \cos^2 \theta = b^2 c^2 \tag{2} \]

### Step 3: Square the second equation
Now, take the second equation and square both sides:
\[ (c \sin \theta)^2 = d^2 \]
This simplifies to:
\[ c^2 \sin^2 \theta = d^2 \tag{3} \]

### Step 4: Multiply by \( a^2 \)
Multiply both sides of equation (3) by \( a^2 \):
\[ a^2 c^2 \sin^2 \theta = a^2 d^2 \tag{4} \]

### Step 5: Add equations (2) and (4)
Now, we add equations (2) and (4):
\[ a^2 c^2 \cos^2 \theta + a^2 c^2 \sin^2 \theta = b^2 c^2 + a^2 d^2 \]

### Step 6: Factor out \( a^2 c^2 \)
On the left side, we can factor out \( a^2 c^2 \):
\[ a^2 c^2 (\cos^2 \theta + \sin^2 \theta) = b^2 c^2 + a^2 d^2 \]

### Step 7: Use the Pythagorean identity
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ a^2 c^2 \cdot 1 = b^2 c^2 + a^2 d^2 \]
This simplifies to:
\[ a^2 c^2 = b^2 c^2 + a^2 d^2 \]

### Conclusion
Thus, the correct relationship is:
\[ a^2 c^2 = b^2 c^2 + a^2 d^2 \]
This corresponds to option 1:
\[ \text{Option 1: } a^2 c^2 = b^2 c^2 + a^2 d^2 \]