If `sin A + sin^2 A + sin^3 A = 1` , then find the value of `cos^6 A-4 cos^4 A + 8 cos^2 A`.

Answer

To solve the equation \( \sin A + \sin^2 A + \sin^3 A = 1 \) and find the value of \( \cos^6 A - 4 \cos^4 A + 8 \cos^2 A \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin A + \sin^2 A + \sin^3 A = 1 \]
We can rearrange this to isolate \( \sin A + \sin^3 A \):
\[ \sin A + \sin^3 A = 1 - \sin^2 A \]
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we have:
\[ \sin A + \sin^3 A = \cos^2 A \]

### Step 2: Square both sides
Now, we square both sides:
\[ (\sin A + \sin^3 A)^2 = \cos^4 A \]
Expanding the left-hand side:
\[ \sin^2 A + 2 \sin A \sin^3 A + \sin^6 A = \cos^4 A \]
This simplifies to:
\[ \sin^2 A + 2 \sin^4 A + \sin^6 A = \cos^4 A \]

### Step 3: Substitute \( \sin^2 A \)
We know that \( \sin^2 A = 1 - \cos^2 A \). Substituting this into the equation:
\[ (1 - \cos^2 A) + 2(1 - \cos^2 A)^2 + (1 - \cos^2 A)^3 = \cos^4 A \]

### Step 4: Expand and simplify
Now we expand and simplify:
1. \( (1 - \cos^2 A)^2 = 1 - 2\cos^2 A + \cos^4 A \)
2. \( (1 - \cos^2 A)^3 = 1 - 3\cos^2 A + 3\cos^4 A - \cos^6 A \)

Substituting these into the equation gives:
\[ 1 - \cos^2 A + 2(1 - 2\cos^2 A + \cos^4 A) + (1 - 3\cos^2 A + 3\cos^4 A - \cos^6 A) = \cos^4 A \]

### Step 5: Combine like terms
Combining all the terms leads to:
\[ 1 - \cos^2 A + 2 - 4\cos^2 A + 2\cos^4 A + 1 - 3\cos^2 A + 3\cos^4 A - \cos^6 A = \cos^4 A \]
This simplifies to:
\[ 4 - 8\cos^2 A + 5\cos^4 A - \cos^6 A = \cos^4 A \]

### Step 6: Rearranging
Rearranging gives:
\[ -\cos^6 A + 4\cos^4 A - 8\cos^2 A + 4 = 0 \]

### Step 7: Substitute \( x = \cos^2 A \)
Let \( x = \cos^2 A \). The equation becomes:
\[ -x^3 + 4x^2 - 8x + 4 = 0 \]
Multiplying through by -1 gives:
\[ x^3 - 4x^2 + 8x - 4 = 0 \]

### Step 8: Factor or use the Rational Root Theorem
By testing possible rational roots, we find that \( x = 2 \) is a root. Thus, we can factor:
\[ (x - 2)(x^2 - 2x + 2) = 0 \]
The quadratic does not have real roots, so we only have \( x = 2 \).

### Step 9: Calculate the value
Now substituting back:
\[ \cos^2 A = 2 \implies \cos^4 A = 4 \quad \text{and} \quad \cos^6 A = 8 \]
Now, substituting these into the expression we need to evaluate:
\[ \cos^6 A - 4\cos^4 A + 8\cos^2 A = 8 - 4(4) + 8(2) \]
Calculating gives:
\[ 8 - 16 + 16 = 8 \]

### Final Answer
Thus, the value of \( \cos^6 A - 4 \cos^4 A + 8 \cos^2 A \) is:
\[ \boxed{8} \]