If `3 sin A+5 cos A =5,` then the value of `(3 cos A-5 sin A )^(2) ` is

Options

A. 4
B. 5
C. 2
D. 9

Answer

To solve the equation \(3 \sin A + 5 \cos A = 5\) and find the value of \((3 \cos A - 5 \sin A)^2\), we can follow these steps:

### Step 1: Square the given equation
Starting with the equation:
\[ 3 \sin A + 5 \cos A = 5 \]
We square both sides:
\[ (3 \sin A + 5 \cos A)^2 = 5^2 \]
This gives us:
\[ 9 \sin^2 A + 30 \sin A \cos A + 25 \cos^2 A = 25 \]

### Step 2: Use the Pythagorean identity
We know that \(\sin^2 A + \cos^2 A = 1\). Thus, we can replace \(9 \sin^2 A + 25 \cos^2 A\) with:
\[ 9 \sin^2 A + 25 \cos^2 A = 9 \sin^2 A + 25(1 - \sin^2 A) = 9 \sin^2 A + 25 - 25 \sin^2 A = -16 \sin^2 A + 25 \]
Substituting this back into our equation gives:
\[ -16 \sin^2 A + 25 + 30 \sin A \cos A = 25 \]
Simplifying this, we find:
\[ -16 \sin^2 A + 30 \sin A \cos A = 0 \]

### Step 3: Factor the equation
Factoring out \(\sin A\):
\[ \sin A(-16 \sin A + 30 \cos A) = 0 \]
This gives us two cases:
1. \(\sin A = 0\)
2. \(-16 \sin A + 30 \cos A = 0\)

### Step 4: Solve the first case
If \(\sin A = 0\), then \(A = n\pi\) for \(n \in \mathbb{Z}\). In this case:
\[ 3 \cos A - 5 \sin A = 3 \cos(n\pi) - 5 \cdot 0 = 3(-1)^n \]
Thus,
\[ (3 \cos A - 5 \sin A)^2 = (3(-1)^n)^2 = 9 \]

### Step 5: Solve the second case
For the second case:
\[ -16 \sin A + 30 \cos A = 0 \implies 30 \cos A = 16 \sin A \implies \tan A = \frac{30}{16} = \frac{15}{8} \]
Using the identity \(\sin^2 A + \cos^2 A = 1\), we can find \(\sin A\) and \(\cos A\):
Let \(\sin A = 15k\) and \(\cos A = 8k\) for some \(k\):
\[ (15k)^2 + (8k)^2 = 1 \implies 225k^2 + 64k^2 = 1 \implies 289k^2 = 1 \implies k^2 = \frac{1}{289} \implies k = \frac{1}{17} \]
Thus:
\[ \sin A = \frac{15}{17}, \quad \cos A = \frac{8}{17} \]
Now substituting back:
\[ 3 \cos A - 5 \sin A = 3 \cdot \frac{8}{17} - 5 \cdot \frac{15}{17} = \frac{24}{17} - \frac{75}{17} = \frac{-51}{17} \]
Calculating the square:
\[ (3 \cos A - 5 \sin A)^2 = \left(\frac{-51}{17}\right)^2 = \frac{2601}{289} = 9 \]

### Conclusion
In both cases, we find that:
\[ (3 \cos A - 5 \sin A)^2 = 9 \]

Thus, the final answer is:
\[ \boxed{9} \]